Suppose a family has 5 children. Suppose the probability of having a girl is 1/2. find the probability that the family has exactly 3 girls and 2 boys?
that would be
C(5,3)(1/2)^3(1/2)^2
= 10/32 = 5/16
C(5,3)(1/2)^3(1/2)^2
= 10/32 = 5/16
To find the probability of a family having exactly 3 girls and 2 boys out of 5 children, we can use the concept of binomial probability.
The probability of having a girl is given as 1/2, which means the probability of having a boy is also 1/2.
Now, let's calculate the probability using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of having exactly k successes
- "n choose k" represents the number of ways to choose k successes out of n trials
- p is the probability of success in one trial
- (1-p) is the probability of failure in one trial
In this case, we have:
- n = 5 (total number of children)
- k = 3 (number of girls)
- p = 1/2 (probability of having a girl)
Let's plug in the values and calculate the probability:
P(X = 3) = (5 choose 3) * (1/2)^3 * (1 - 1/2)^(5 - 3)
Using the binomial coefficient formula (n choose k):
(5 choose 3) = 5! / (3! * (5-3)!) = (5 * 4) / (2 * 1) = 10
Now, let's substitute the values into the probability formula:
P(X = 3) = 10 * (1/2)^3 * (1/2)^2
= 10 * (1/8) * (1/4)
= 10 * 1/32
= 10/32
= 5/16
Therefore, the probability that the family has exactly 3 girls and 2 boys out of 5 children is 5/16.
To find the probability of having exactly 3 girls and 2 boys in a family with 5 children, we can use the concept of a binomial distribution. In this case, the probability of having a girl is given as 1/2.
The binomial distribution formula can be used to calculate the probability of obtaining a specific number of successes (girls) in a fixed number of trials (children), assuming each trial is independent and has the same probability of success.
The formula for the probability of exactly k successes (girls) in n trials (children) is:
P(k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- (n choose k) represents the number of ways to choose k successes from n trials and can be calculated as n! / (k!(n-k)!)
- p is the probability of success (having a girl)
- (1-p) is the probability of failure (having a boy)
- n is the total number of trials (children)
- k is the number of successful trials (girls)
In this specific example, we want to find the probability of having exactly 3 girls (k = 3) and 2 boys (n-k = 2) in a family with 5 children (n = 5):
P(3 girls) = (5 choose 3) * (1/2)^3 * (1-(1/2))^(5-3)
Now, let's calculate the probability step by step:
First, calculate the binomial coefficient (5 choose 3) = 5! / (3!(5-3)!) = 5! / (3!2!) = 10.
Next, substitute the values into the formula:
P(3 girls) = 10 * (1/2)^3 * (1/2)^2
= 10 * (1/8) * (1/4)
= 10/32
= 5/16
Therefore, the probability that the family has exactly 3 girls and 2 boys is 5/16.