The probability distribution is:
P(X=0)=0.0625
P(X=1)=0.5
P(X=2)=0.375
P(X=3)=0.25
P(X=4)=0.0625
In a family of four children where a girl is as likely as a boy, the possible values for the number of boys (random variable X) can range from 0 to 4. We can use the binomial probability distribution to find the probabilities.
The probability mass function (PMF) for a binomial distribution is given by:
P(X=k)=( n k )p^k (1−p)^n−k
where:
n is the number of trials (number of children in this case),
k is the number of successes (number of boys in this case
p is the probability of success on each trial (probability of having a boy),
(n k) is the binomial coefficient, which represents the number of ways to choose
k successes from
n trials.
In this case,
n=4 (four children), and
p=0.5 (equal probability of having a boy or a girl).
Now, let's calculate the probabilities for each value of X:
P(X=0)=( 0 4 )(0.5)^0 (0.5)^4
P(X=1)=( 1 4 )(0.5)^1 (0.5)^3
P(X=2)=( 2 4 )(0.5)^2 (0.5)^2
P(X=3)=( 3 4 )(0.5)^3 (0.5)^1
P(X=4)=( 4 4 )(0.5)^4 (0.5)^0
Let's calculate these probabilities:
P(X=0)=( 0 4 )⋅1⋅0.0625=1⋅0.0625=0.0625
P(X=1)=( 1 4 )⋅0.5⋅0.125=4⋅0.125=0.5
P(X=2)=( 2 4 )⋅0.25⋅0.25=6⋅0.0625=0.375
P(X=3)=( 3 4 )⋅0.125⋅0.5=4⋅0.0625=0.25
P(X=4)=( 4 4 )⋅0.0625⋅1=1⋅0.0625=0.0625
So, the probability distribution is:
P(X=0)=0.0625
P(X=1)=0.5
P(X=2)=0.375
P(X=3)=0.25
P(X=4)=0.0625