It has been found out that the probability that a child is a male in family is 0.4. In a family of 6 children, what is the probability that; i. exactly 5 are boys ? ii. at least 4 are boys ? iii. all boys
prob(male) = .4
prob(female) = .6
prob(5 of 6 are male)
= C(6,5) (.4)^5 (.6) = ...
prob(at least 4 of the 6 are boys)
= prob(4 boys) + prob(5 boys) + prob(6 boy s)
= C(6,4) (.4)^4 (.6)^2 + C(6,5) (.4)^5 (.6) + .4^6
= ---
all boys = .4^6
To calculate the probabilities, we can use the concept of binomial probability. The binomial probability formula is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successful outcomes.
n is the number of trials.
k is the number of successful outcomes.
p is the probability of a single successful outcome.
(nCk) represents the number of combinations of n items taken k at a time.
Let's calculate the probabilities for each case:
i. Probability that exactly 5 are boys:
Using the binomial probability formula, we have:
P(X = 5) = (6C5) * (0.4)^5 * (1-0.4)^(6-5)
Calculating it step by step:
(6C5) = 6 (since there is only one way for 5 boys out of 6 children)
(0.4)^5 = 0.01024
(1-0.4)^(6-5) = 0.6^1 = 0.6
P(X = 5) = 6 * 0.01024 * 0.6 = 0.036864
Therefore, the probability that exactly 5 children are boys is approximately 0.036864 or 3.6864%.
ii. Probability that at least 4 are boys:
This can be calculated by finding the probabilities of having 4, 5, or 6 boys and adding them together.
P(at least 4 boys) = P(X = 4) + P(X = 5) + P(X = 6)
We have already calculated P(X = 5), so let's calculate P(X = 4) and P(X = 6):
P(X = 4) = (6C4) * (0.4)^4 * (1-0.4)^(6-4)
Calculating:
(6C4) = 15
(0.4)^4 = 0.0256
(1-0.4)^(6-4) = 0.6^2 = 0.36
P(X = 4) = 15 * 0.0256 * 0.36 = 0.13824
P(X = 6) = (6C6) * (0.4)^6 * (1-0.4)^(6-6)
Calculating:
(6C6) = 1
(0.4)^6 = 0.004096
(1-0.4)^(6-6) = 0.6^0 = 1
P(X = 6) = 1 * 0.004096 * 1 = 0.004096
Adding them together:
P(at least 4 boys) = 0.13824 + 0.036864 + 0.004096 = 0.1792
Therefore, the probability that at least 4 children are boys is approximately 0.1792 or 17.92%.
iii. Probability that all children are boys:
This is the probability of having 6 boys out of 6 children, which can be calculated using the binomial probability formula:
P(X = 6) = (6C6) * (0.4)^6 * (1-0.4)^(6-6)
Calculating:
(6C6) = 1 (only one combination for 6 boys out of 6 children)
(0.4)^6 = 0.004096
(1-0.4)^(6-6) = 0.6^0 = 1
P(X = 6) = 1 * 0.004096 * 1 = 0.004096
Therefore, the probability that all children are boys is approximately 0.004096 or 0.4096%.