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Should You Vaccinate Your Child Against Measles? Understanding the Math Behind the Decision

December 27, 2025
16 min read
By Avi Turetsky

Using Bayes' Theorem to understand vaccine risk assessment. The current low rate of measles exists because of vaccination—but what risk does your unvaccinated child actually face?

You're sitting in your pediatrician's office, holding your one-year-old. The doctor recommends the MMR vaccine—measles, mumps, and rubella. You've heard measles is rare in the United States. You've also heard stories about vaccine reactions. Both risks feel real. Both feel frightening.

This is one of the hardest decisions parents face. You want to protect your child, but the information landscape feels overwhelming and contradictory. Let's cut through the noise and look at the actual numbers using Bayes' Theorem—a mathematical framework for thinking clearly about risk.

This isn't about judging anyone's choices. It's about understanding why our intuitions about vaccine risk often lead us astray, and what the math actually tells us.

What the Bayesian analysis reveals:

In a school where 85% of children are vaccinated, 80% of serious health complications happen to the 15% who are unvaccinated. Even vaccinated children face 3× higher risk in low-vaccination schools compared to high-vaccination schools. The math shows that vaccination protects your child—and everyone benefits when community immunity is strong.

The Paradox Parents Face

Here's the dilemma: Measles feels distant. As of December 2025, there were 2,012 confirmed measles cases in the United States—out of a population of 335 million people.[1] That's roughly 1 in 166,000 people. It's natural to think: "My child probably won't encounter measles, so why risk vaccine side effects?"

Meanwhile, vaccine side effects—fever, rash, soreness—are common and visible. About 1 in 6 children will develop a fever after the MMR vaccine.[2] You can see it happen. You can feel your child's discomfort.

The invisible becomes abstract. The visible becomes concrete.

This is where our intuition fails us. We're comparing the current low rate of measles—which exists because of high vaccination rates—to the risk our unvaccinated child would actually face. That's the wrong comparison.

The Numbers You Need to Know

Before we dive into the Bayesian analysis, let's establish the key facts from authoritative medical sources.

Measles: One of the Most Contagious Diseases

Measles has a basic reproduction number (R₀) of 12-18, meaning each infected person will infect 12-18 others in an unvaccinated population.[3] This makes measles one of the most contagious diseases known to medicine—far more contagious than COVID-19 (R₀ ~2-3) or influenza (R₀ ~1-2).

Measles spreads through the air. You can catch it just by being in a room where an infected person coughed or sneezed—even after they've left.[4] Among unvaccinated people living in close contact with an infected person, the attack rate is approximately 90%.[5]

Measles Complications Are Common and Serious

Measles isn't "just a rash." Here are the complication rates from the CDC:[6]

ComplicationRateNotes
Hospitalization1 in 5 (20%)Among unvaccinated people who get measles
Ear infections1 in 10 (10%)Common in children
Pneumonia1 in 20 (5%)Most common cause of death in young children
Encephalitis1 in 1,000 (0.1%)Brain swelling; can cause deafness or intellectual disability
Death1-3 in 1,000 (0.1-0.3%)From respiratory and neurologic complications
SSPE7-11 in 100,000Rare but fatal brain disease developing 7-10 years after infection

Key insight: If your child gets measles, there's a 1 in 5 chance they'll need hospitalization. That's not a rare complication—that's a common outcome.

MMR Vaccine: Highly Effective and Safe

The MMR vaccine has been used since 1971 and has an exceptional safety and effectiveness record:

Effectiveness:[7]

  • One dose: 93% effective against measles
  • Two doses: 97% effective against measles

Understanding Vaccine Side Effects

Let's be honest: vaccine side effects are real, and it's completely reasonable for parents to worry about them. When your child develops a fever or feels unwell after a vaccine, that discomfort is visible and immediate. It's natural to wonder whether you made the right choice.

The key question isn't whether side effects exist—they do. The question is: How do vaccine risks compare to disease risks? Let's look at the actual numbers:

Common side effects (mild and temporary):[2]

  • Sore arm from the shot (very common)
  • Fever: ~1 in 6 children (17%)
  • Mild rash: ~1 in 20 children (5%)
  • Temporary joint pain (mostly in adult women)

Rare but more serious side effects:

  • Febrile seizures: ~1 in 3,000-4,000 children (0.025-0.033%)
  • Immune thrombocytopenic purpura (temporary low platelet count): ~1 in 40,000 (0.0025%)
  • Severe allergic reaction (anaphylaxis): Extremely rare (~1 in 1,000,000)

Important context about febrile seizures: While frightening to witness, febrile seizures from the MMR vaccine are not associated with long-term neurological damage or epilepsy. They're a temporary reaction to fever and resolve without lasting effects.

The CDC's bottom line:

"Most people who get MMR vaccine do not have any serious problems with it. Getting MMR vaccine is much safer than getting measles, mumps, or rubella."[2]

The Bayesian Analysis: Who Bears the Risk?

Now let's use Bayes' Theorem to understand the true distribution of risk. We'll calculate the probability of serious complications for both vaccinated and unvaccinated children in different school environments, then use Bayesian inversion to see who actually bears the burden of health complications.

This is where most people's intuition goes wrong. They look at the current rate (2,012 cases out of 335 million people) and think that's their child's risk. But that rate exists in a population where 92.7% of kindergartners are vaccinated.[1] More importantly, your child's risk depends on their school's vaccination rate, not the national average.

The Critical Insight: School-Level Risk

A 2025 study of the West Texas measles outbreak (762 cases, 99 hospitalizations, 2 deaths) revealed that 93% of patients were unvaccinated.[11] County-level data masked dramatic school-level variation—individual schools ranged from 46% to 97% vaccinated even in counties with high overall rates.

The key finding:

Modeling shows that introducing even a single measles case into a school with 85% vaccination coverage is more likely than not (>50%) to trigger an outbreak.[11]

This means your child's risk isn't determined by national statistics—it's determined by where they go to school. Let's use Bayes' Theorem to calculate the risk for three different scenarios.

The Bayesian Question

Here's where Bayes' Theorem comes in. Instead of asking "What's my child's risk if I vaccinate or don't vaccinate?", let's flip the question:

"If a child in this school experiences a serious complication, what's the probability they were vaccinated vs. unvaccinated?"

This is genuine Bayesian reasoning—we're inverting a conditional probability. We know P(complication | vaccinated) and P(complication | unvaccinated), but we want to calculate P(vaccinated | complication).

Why does this matter? Parents often worry: "What if my child is the one who has a bad reaction to the vaccine?" This inverted question shows: "If a child has a serious health problem, was it more likely from getting the vaccine or skipping it?"

Bayes' Theorem Formula:

P(vaccinated | complication) = [P(complication | vaccinated) × P(vaccinated)] / P(complication)

Where:

  • P(vaccinated) = the school's vaccination rate (our prior)
  • P(complication | vaccinated) = risk if vaccinated (from vaccine side effects + rare breakthrough infections)
  • P(complication | unvaccinated) = risk if unvaccinated (from measles exposure)
  • P(complication) = total probability of complications across all children = P(complication | vaccinated) × P(vaccinated) + P(complication | unvaccinated) × P(unvaccinated)

Let's calculate this for three different school scenarios. You'll see that even in schools where most children are vaccinated, the vast majority of complications happen to unvaccinated children.

Scenario A: High-Vaccination School (95%+ Vaccinated)

For an Unvaccinated Child:

In a school with strong herd immunity:

  • Probability of exposure to measles over 18 years: ~1-2% (outbreaks are rare)
  • Probability of infection given exposure: 90% (well-established attack rate)[5]
  • Probability of getting measles: 1.5% × 90% = 1.35%

If they get measles:

  • Probability of hospitalization: 1.35% × 20% = 0.27% (1 in 370)
  • Probability of death: 1.35% × 0.2% = 0.0027% (1 in 37,000)

For a Vaccinated Child:

Vaccinated children also benefit from herd immunity! In a 95% vaccinated school:

  • Probability of exposure: ~1.5% (same as unvaccinated—herd immunity reduces circulating measles)
  • Probability of infection given exposure: 3% (vaccine is 97% effective)
  • Probability of getting measles: 1.5% × 3% = 0.045%
  • Probability of hospitalization from measles: 0.045% × 20% = 0.009%
  • Probability of febrile seizure from vaccine: 0.03% (1 in 3,000—no lasting harm)
  • Total serious complication risk: ~0.04% (vaccine side effects + rare breakthrough infections)

The Bayesian Calculation for Scenario A:

Now let's apply Bayes' Theorem:

  • P(vaccinated) = 95% (the school vaccination rate)
  • P(complication | vaccinated) = 0.04% (vaccine side effects + breakthrough infections)
  • P(complication | unvaccinated) = 0.27% (measles hospitalization)

P(complication) = (0.0004 × 0.95) + (0.0027 × 0.05) = 0.00038 + 0.000135 = 0.051%

P(vaccinated | complication) = (0.0004 × 0.95) / 0.00051 = 73.5%

Scenario A Result:

Even in a school where 95% of children are vaccinated, only 73.5% of complications come from vaccinated children. The remaining 26.5% of complications come from the 5% who are unvaccinated—meaning unvaccinated children are 5.3× more likely to experience serious complications than vaccinated children.

Scenario B: Medium-Vaccination School (85% Vaccinated)

For an Unvaccinated Child:

Below the 95% herd immunity threshold, outbreaks become likely:

  • Probability of exposure over 18 years: ~15% (outbreaks are common)
  • Probability of infection given exposure: 90%
  • Probability of getting measles: 15% × 90% = 13.5% (1 in 7 children)

If they get measles:

  • Probability of hospitalization: 13.5% × 20% = 2.7% (1 in 37)
  • Probability of death: 13.5% × 0.2% = 0.027% (1 in 3,700)

For a Vaccinated Child:

Even vaccinated children face higher risk in lower-vaccination schools:

  • Probability of exposure: ~15% (more circulating measles than Scenario A)
  • Probability of infection given exposure: 3% (vaccine is 97% effective)
  • Probability of getting measles: 15% × 3% = 0.45%
  • Probability of hospitalization from measles: 0.45% × 20% = 0.09%
  • Probability of febrile seizure from vaccine: 0.03%
  • Total serious complication risk: ~0.12% (3× higher than Scenario A!)

The Bayesian Calculation for Scenario B:

Now let's apply Bayes' Theorem:

  • P(vaccinated) = 85% (the school vaccination rate)
  • P(complication | vaccinated) = 0.12% (vaccine side effects + breakthrough infections)
  • P(complication | unvaccinated) = 2.7% (measles hospitalization)

P(complication) = (0.0012 × 0.85) + (0.027 × 0.15) = 0.00102 + 0.00405 = 0.507%

P(vaccinated | complication) = (0.0012 × 0.85) / 0.00507 = 20.1%

Scenario B Result:

In a school where 85% of children are vaccinated, only 20% of complications come from vaccinated children. The remaining 80% of complications come from the 15% who are unvaccinated. This is the power of Bayes' Theorem: it shows that even though unvaccinated children are a minority, they bear the vast majority of the health burden—and vaccinated children face 3× higher risk than in high-vaccination schools.

Scenario C: Low-Vaccination School (<80% Vaccinated)

For an Unvaccinated Child:

Well below herd immunity, approaching pre-vaccine era conditions:

  • Probability of exposure over 18 years: ~35% (outbreaks are highly likely)
  • Probability of infection given exposure: 90%
  • Probability of getting measles: 35% × 90% = 31.5% (nearly 1 in 3 children)

If they get measles:

  • Probability of hospitalization: 31.5% × 20% = 6.3% (1 in 16)
  • Probability of death: 31.5% × 0.2% = 0.063% (1 in 1,587)

For a Vaccinated Child:

In low-vaccination schools, even vaccinated children face dramatically elevated risk:

  • Probability of exposure: ~35% (widespread measles circulation)
  • Probability of infection given exposure: 3% (vaccine is 97% effective)
  • Probability of getting measles: 35% × 3% = 1.05%
  • Probability of hospitalization from measles: 1.05% × 20% = 0.21%
  • Probability of febrile seizure from vaccine: 0.03%
  • Total serious complication risk: ~0.25% (6× higher than Scenario A!)

The Bayesian Calculation for Scenario C:

Now let's apply Bayes' Theorem:

  • P(vaccinated) = 75% (the school vaccination rate)
  • P(complication | vaccinated) = 0.25% (vaccine side effects + breakthrough infections)
  • P(complication | unvaccinated) = 6.3% (measles hospitalization)

P(complication) = (0.0025 × 0.75) + (0.063 × 0.25) = 0.001875 + 0.01575 = 1.76%

P(vaccinated | complication) = (0.0025 × 0.75) / 0.0176 = 10.7%

Scenario C Result:

In a school where 75% of children are vaccinated, only 10.7% of complications come from vaccinated children. An astonishing 89.3% of complications come from the 25% who are unvaccinated. In low-vaccination environments, the unvaccinated minority bears almost the entire health burden—and vaccinated children face 6× higher risk than in high-vaccination schools.

Summary: Risk by School Vaccination Rate

School Vaccination RateChild StatusRisk of HospitalizationRisk of Death
95%+ (High)Unvaccinated0.27% (1 in 370)0.0027% (1 in 37,000)
Vaccinated0.04% (1 in 2,500)0.0001% (1 in 1 million)
85% (Medium)Unvaccinated2.7% (1 in 37)0.027% (1 in 3,700)
Vaccinated0.12% (1 in 833)0.0002% (1 in 500,000)
<80% (Low)Unvaccinated6.3% (1 in 16)0.063% (1 in 1,587)
Vaccinated0.25% (1 in 400)0.0005% (1 in 200,000)

The Free Rider Problem

Notice something crucial in the table above: Both vaccinated and unvaccinated children benefit from herd immunity. Vaccinated child risk increases from 0.04% to 0.25% as school vaccination drops from 95% to 75%. Unvaccinated child risk increases even more dramatically: from 0.27% to 6.3%.

If you choose not to vaccinate your child, their safety depends entirely on other parents vaccinating their children. In a high-vaccination school, your unvaccinated child benefits from herd immunity—they're "free riding" on community protection. But this protection is fragile, and even vaccinated children pay a price when community immunity erodes.

The paradox:

The safer it feels to skip vaccination (because measles seems rare), the more that safety depends on others vaccinating. As more parents make the same choice, herd immunity breaks down, and all unvaccinated children face dramatically higher risk.

This is exactly what happened in West Texas. As vaccination rates declined in certain schools and communities, the protection disappeared. The 2025 outbreak resulted in 762 cases—a preventable tragedy that claimed lives and hospitalized nearly 100 children.

The Bayesian Conclusion

Bayes' Theorem teaches us to update our beliefs based on evidence. When we look at the evidence clearly:

  • Measles is one of the most contagious and dangerous childhood diseases
  • The MMR vaccine is one of the safest and most effective vaccines ever developed
  • Unvaccinated children bear nearly all the risk as vaccination rates decline
  • The math isn't close—vaccination is safer by orders of magnitude

As a parent, you're trying to protect your child. That's rational. That's loving. But protecting your child means making decisions based on evidence, not fear. The Bayesian approach asks: Given all the evidence, what's the most likely outcome? And the answer is clear: vaccinating your child gives them the best chance of staying healthy and safe.

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About the Author

Written by Avi Turetsky

Passionate about making Bayesian reasoning accessible and helping people make better decisions through probabilistic thinking.

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References

  1. CDC. "Measles Cases and Outbreaks." December 23, 2025. https://www.cdc.gov/measles/data-research/index.html
  2. CDC. "Measles, Mumps, Rubella (MMR) Vaccine Safety." July 31, 2024. https://www.cdc.gov/vaccine-safety/vaccines/mmr.html
  3. Guerra FM, et al. "The basic reproduction number (R₀) of measles: a systematic review." The Lancet Infectious Diseases, 2017. https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(17)30307-9/abstract
  4. CDC. "About Measles." May 9, 2024. https://www.cdc.gov/measles/about/index.html
  5. Gambrell A, et al. "Estimating the number of US children susceptible to measles." PMC, 2022. https://pmc.ncbi.nlm.nih.gov/articles/PMC9197781/
  6. CDC. "Measles Symptoms and Complications." May 9, 2024. https://www.cdc.gov/measles/signs-symptoms/index.html
  7. CDC. "Measles Vaccination." January 17, 2025. https://www.cdc.gov/measles/vaccines/index.html
  8. CDC. "Measles — United States, January 1, 2020–March 28, 2024." MMWR, 2024. https://www.cdc.gov/mmwr/volumes/73/wr/mm7314a1.htm
  9. CDC. "History of Measles." February 5, 2025. https://www.cdc.gov/measles/about/history.html
  10. CDC. "Vaccines and Autism." https://www.cdc.gov/vaccine-safety/vaccines/mmr.html

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