The Smith family was one of the first to come to the U.S. They had 8 children. Assuming that the probability of a child being a girl is .5, find the probability that the Smith family had:
a. at least 6 girls
b. at most 5 girls
Use binomial distribution because:
1. Bernoulli trials (either girl or boy)
2. number of trials (n=9) known.
3. probability (p=0.5) of outcome known and remains constant throughout trials
4. trials are independent of each other.
n=9
p=0.5
P(X=k)=C(n,k)*p^k*(1-p)^(n-k)
where
C(n,k) is the number of combinations of k objects taken from n.
At least 5 girls means X=5,6,7,8,9
So
P(X>=5)
=P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)
=0.24609+0.16406+0.07031+0.01758+0.00195
=0.5000
To find the probability that the Smith family had a certain number of girls, we can use the concept of binomial probability. Binomial probability is used to calculate the probability of getting a certain number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success.
In this case, each child being a girl or a boy can be considered a Bernoulli trial, as each child has a 50% chance of being a girl. The number of girls the Smith family had follows a binomial distribution with n = 8 (number of children) and p = 0.5 (probability of a child being a girl).
a. To find the probability that the Smith family had at least 6 girls, we need to calculate the probability of having 6, 7, or 8 girls. We can use the binomial probability formula and compute these individual probabilities:
P(X = 6) = C(8,6) * (0.5)^6 * (1-0.5)^2
P(X = 7) = C(8,7) * (0.5)^7 * (1-0.5)^1
P(X = 8) = C(8,8) * (0.5)^8 * (1-0.5)^0
P(at least 6 girls) = P(X = 6) + P(X = 7) + P(X = 8)
For each calculation, C(n, k) represents the number of combinations of choosing k successes from n trials, and (0.5)^k represents the probability of k successes. (1-0.5)^(n-k) represents the probability of (n-k) failures.
b. To find the probability that the Smith family had at most 5 girls, we need to calculate the probability of having 0, 1, 2, 3, 4, or 5 girls. Similarly, we can use the binomial probability formula to compute these individual probabilities:
P(X = 0) = C(8,0) * (0.5)^0 * (1-0.5)^8
P(X = 1) = C(8,1) * (0.5)^1 * (1-0.5)^7
P(X = 2) = C(8,2) * (0.5)^2 * (1-0.5)^6
P(X = 3) = C(8,3) * (0.5)^3 * (1-0.5)^5
P(X = 4) = C(8,4) * (0.5)^4 * (1-0.5)^4
P(X = 5) = C(8,5) * (0.5)^5 * (1-0.5)^3
P(at most 5 girls) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
To calculate the values C(n, k), you can use the combination formula:
C(n, k) = n! / (k! * (n-k)!)
where "!" represents the factorial of a number.
Now you can use a calculator or a spreadsheet to compute the individual probabilities and add them up to find the final probabilities.