What is the expected number of female children in a family with three children?
Answer (1)
Define X as the number of female children, which can be 0, 1, 2, or 3.
Calculate the probability for each value of X using the binomial probability formula.
Compute the expected value E ( X ) by summing the product of each value and its probability.
The expected number of female children is 1.5 .
Explanation
Analyze the problem Let's analyze the problem. We are asked to find the expected number of female children in a family with three children. We assume that each child is equally likely to be male or female, meaning the probability of having a female child is 0.5, and the genders of the children are independent.
Define random variable and binomial probability Let X be the random variable representing the number of female children. X can take values 0, 1, 2, or 3. We need to calculate the probability of each of these values. We can use the binomial probability formula: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k where n is the number of trials (children), k is the number of successes (female children), and p is the probability of success (probability of having a female child). In our case, n = 3 and p = 0.5 .
Calculate probabilities Now, let's calculate the probabilities for each possible value of X :
For X = 0 :
P ( X = 0 ) = ( 0 3 ) ( 0.5 ) 0 ( 0.5 ) 3 − 0 = 1 ∗ 1 ∗ ( 0.5 ) 3 = 0.125
For X = 1 :
P ( X = 1 ) = ( 1 3 ) ( 0.5 ) 1 ( 0.5 ) 3 − 1 = 3 ∗ 0.5 ∗ ( 0.5 ) 2 = 3 ∗ 0.5 ∗ 0.25 = 0.375
For X = 2 :
P ( X = 2 ) = ( 2 3 ) ( 0.5 ) 2 ( 0.5 ) 3 − 2 = 3 ∗ ( 0.5 ) 2 ∗ 0.5 = 3 ∗ 0.25 ∗ 0.5 = 0.375
For X = 3 :
P ( X = 3 ) = ( 3 3 ) ( 0.5 ) 3 ( 0.5 ) 3 − 3 = 1 ∗ ( 0.5 ) 3 ∗ 1 = 0.125
Calculate expected value Now we calculate the expected value of X using the formula: E ( X ) = k = 0 ∑ n k ∗ P ( X = k )
E ( X ) = 0 ∗ P ( X = 0 ) + 1 ∗ P ( X = 1 ) + 2 ∗ P ( X = 2 ) + 3 ∗ P ( X = 3 ) E ( X ) = 0 ∗ 0.125 + 1 ∗ 0.375 + 2 ∗ 0.375 + 3 ∗ 0.125 = 0 + 0.375 + 0.75 + 0.375 = 1.5
State the final answer Therefore, the expected number of female children in a family with three children is 1.5.
Examples
This concept is useful in genetics to predict the average number of offspring with a particular trait in a population. For instance, if you're studying a genetic disorder that affects males and females equally, you can use expected value to estimate how many children in a family with a certain number of kids are likely to inherit the disorder.
