Ignoring twins and other mutiple births, assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5. Referring to the information above, the probability that at least one of the next three babies are of the same sex is...?
I don't know how to approach this problem. Please help!
Is this a trick question? "One" cannot be the same sex. It takes at least two to be the same.
I hope this helps a little more. Thanks for asking.
For these types of problems, I like drawing a probability tree.
That said, I really don't understand your question. With 3 babies, at least 2 will be of the same sex (BBG or BGG) possibly all 3 are the same sex (BBB or GGG). Could you clarify?
To find the probability that at least one of the next three babies is of the same sex, we can use the complement rule.
First, let's calculate the probability that all three babies are of different sexes. Since the probability of a baby being a boy or a girl is both 0.5, the probability that the first baby is a boy and the second baby is a girl or vice versa is:
0.5 * 0.5 = 0.25
Since there are two possible ways to have different sexes for the first two babies (boy-girl or girl-boy), we multiply the above result by 2:
0.25 * 2 = 0.5
Now, the probability that the third baby has a different sex than the first two is also 0.5.
Therefore, the probability that all three babies have different sexes is:
0.5 * 0.5 * 0.5 = 0.125
Now, to find the probability that at least one baby has the same sex, we subtract the probability that all three babies have different sexes from 1:
1 - 0.125 = 0.875
Therefore, the probability that at least one of the next three babies is of the same sex is 0.875.
To find the probability that at least one of the next three babies is of the same sex, we can use the concept of complementary probability. Instead of directly calculating the probability that at least one of the next three babies is of the same sex, we'll calculate the probability that none of the next three babies is of the same sex, and then subtract that from 1.
The probability that the first two babies are of different sexes is (0.5 * 0.5) = 0.25. Now, let's consider the third baby. There are two possibilities:
1) It is of the same sex as the first baby, which has a probability of 0.5.
2) It is of the same sex as the second baby, which has a probability of 0.5.
Since these two cases are mutually exclusive, we can add their probabilities: 0.5 + 0.5 = 1.
To find the probability that none of the next three babies is of the same sex, we multiply the probability of the first two babies being of different sexes (0.25) by the probability of all three babies having the opposite sex as the third baby (1). The result is: 0.25 * 1 = 0.25.
Now, to find the probability that at least one of the next three babies is of the same sex, we subtract the probability that none of them are: 1 - 0.25 = 0.75.
Therefore, the probability that at least one of the next three babies is of the same sex is 0.75 or 75%.