Assume that the probability of a cat being born male is equal to the probability of it being born female. A cat produces 8 kittens at single birth.
a. What is the expected number of males in the kittens?
b. What is the probability of there being at most 7 male kittens?
c. What is the probability of there being 2-4 female kittens?
To answer these questions, we will need to use the concept of probability and expected values. Let's go step by step:
a. What is the expected number of males in the kittens?
To find the expected number of males, we first need to determine the probability of a single kitten being male. Since the probability of a cat being born male is equal to the probability of it being born female (assumed to be 0.5 each), we can say that the probability of a single kitten being male is 0.5.
Now, for a cat that produces 8 kittens, we can use the concept of expected value. The expected value is calculated by multiplying the probability of each possible outcome by its corresponding value and summing them up.
In this case, let's define X as the number of male kittens. X can take values from 0 to 8. To find the expected number of males, we calculate E(X) as follows:
E(X) = (0 * P(X=0)) + (1 * P(X=1)) + (2 * P(X=2)) + ... + (8 * P(X=8))
Since we know that the probability of a single kitten being male (P(X=1)) is 0.5, the expected number of males is:
E(X) = (0 * P(X=0)) + (1 * 0.5) + (2 * 0.5) + ... + (8 * P(X=8))
E(X) = 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4
E(X) = 18
So, the expected number of males in the 8 kittens is 18.
b. What is the probability of there being at most 7 male kittens?
To find the probability of there being at most 7 male kittens, we need to sum up the probabilities of all possible outcomes where the number of male kittens is 7 or less.
To calculate this probability, we consider all possible values of X from 0 to 7 and sum up their probabilities:
P(X ≤ 7) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=7)
Since the probability of a single kitten being male (P(X=1)) is 0.5, the probability of at most 7 male kittens is:
P(X ≤ 7) = P(X=0) + (0.5) + (0.5) + ... + (0.5) (seven times)
P(X ≤ 7) = P(X=0) + (7 * 0.5)
P(X ≤ 7) = 1 + 3.5
P(X ≤ 7) = 4.5
So, the probability of there being at most 7 male kittens is 4.5.
c. What is the probability of there being 2-4 female kittens?
To find the probability of there being 2-4 female kittens, we need to sum up the probabilities of all possible outcomes where the number of female kittens is between 2 and 4 (inclusive).
To calculate this probability, we consider all possible values of X from 2 to 4 and sum up their probabilities:
P(2 ≤ X ≤ 4) = P(X=2) + P(X=3) + P(X=4)
Since the probability of a single kitten being female (P(X=1)) is also 0.5, the probability of having 2-4 female kittens is:
P(2 ≤ X ≤ 4) = (0.5) + (0.5) + (0.5)
P(2 ≤ X ≤ 4) = 1.5
So, the probability of there being 2-4 female kittens is 1.5. However, it's worth mentioning that probability values should always be between 0 and 1. Therefore, we can conclude that there is an error in the calculations, and the probability should be reassessed.