Researchers at the University of Pennsylvania School of Medicine have determined that children under 2 years old who sleep with the lights on have a 36% chance of becoming myopic before they turn 16. Children who sleep in darkness have a 21% chance of becoming myopic before they turn 16. A survey indicates that 28% of children under 2 sleep with some light on. If a randomly selected child under 16 is myopic, what is the probability that he/she slept with the light on before they turned 2?

P(M|L)=.36

P(M|L^c)=.21
P(L)=.28
P(L^c)=.72
P(M|L)P(L)+P(M|L^c)P(L^c)
.36*.28+.21*.72
.252
Baye's Rule

m/p=236

To solve this problem, we can use Bayes' theorem. Bayes' theorem states that the probability of an event given another event can be calculated using the following formula:

P(A|B) = (P(B|A) * P(A)) / P(B)

In this case, we want to find the probability that a randomly selected myopic child under 16 slept with the light on before they turned 2. Let's assign the following variables:

A = a randomly selected child under 16 slept with the light on before they turned 2.
B = a randomly selected child under 16 is myopic.

We are given the following information:

P(A) = 28% = 0.28 (the survey indicates that 28% of children under 2 sleep with some light on)
P(B|A) = 36% = 0.36 (children under 2 who sleep with the lights on have a 36% chance of becoming myopic before they turn 16)
P(B) = ? (the probability that a randomly selected child under 16 is myopic)

To find P(B), we need to consider the chance that a randomly selected child under 16 is myopic. This information is not provided in the given problem, so we cannot calculate it. Without this information, we cannot determine the probability that a myopic child slept with the light on before they turned 2.

To find the probability that a randomly selected myopic child slept with the light on before turning 2, we can use Bayes' theorem:

P(Light on | Myopic) = (P(Myopic | Light on) * P(Light on)) / P(Myopic)

Let's break down these terms:

P(Light on | Myopic) represents the probability that a child slept with the light on given that he/she is myopic.
P(Myopic | Light on) represents the probability that a child is myopic given that he/she slept with the light on.
P(Light on) represents the probability that a randomly selected child slept with the light on before turning 2.
P(Myopic) represents the probability that a randomly selected child is myopic.

According to the given information:
P(Light on) = 28% = 0.28
P(Myopic) = (P(Myopic | Light on) * P(Light on)) + (P(Myopic | Darkness) * P(Darkness))
= (36% * 0.28) + (21% * (1 - 0.28))
= 0.1008 + 0.147
= 0.2478

Now, we need to calculate P(Myopic | Light on). We can use Bayes' theorem again:

P(Myopic | Light on) = (P(Light on | Myopic) * P(Myopic)) / P(Light on)

We already know:
P(Light on | Myopic) = 36% = 0.36
P(Myopic) = 0.2478
P(Light on) = 0.28

Calculating:
P(Myopic | Light on) = (0.36 * 0.2478) / 0.28
≈ 0.3176

Therefore, the probability that a randomly selected myopic child slept with the light on before turning 2 is approximately 0.3176 or 31.76%.