A family consists of ten children. Assuming that the sexes are equally probable,
find the expected number of boys in the family.
What is the probability that the expected number of boys will not occur?
(Not sure where to start) I got partial credit and I answered 5 (Question 1) and 1/2 (Question 2)
the answer will be Q1 (5)
Q2 (0.754)
Good Luck :)
How is question 2= 0.754?
The second question is 0.754 because of this equation:
(10C5)= (0.5^5)(0.5)^5
= 0.246
P(E')= 1-P(E)
= 1-0.246
= 0.754
To find the expected number of boys in the family, you need to calculate the probability of each possible outcome and multiply it by the number of boys for that outcome. Since the sexes are equally probable, the probability of having a boy or a girl is 1/2.
Start by determining the number of boys in each possible outcome:
- If there are 0 boys and 10 girls, the probability is (1/2)^10.
- If there is 1 boy and 9 girls, the probability is (1/2)^1 * (1/2)^9.
- If there are 2 boys and 8 girls, the probability is (1/2)^2 * (1/2)^8.
- Similarly, you can calculate the probabilities for 3 boys, 4 boys, and so on, up to 10 boys.
Now, multiply each probability by the respective number of boys in that outcome:
Expected number of boys = 0*(1/2)^10 + 1*(1/2)^1 * (1/2)^9 + 2*(1/2)^2 * (1/2)^8 + ... + 10*(1/2)^10
Expected number of boys = (0 + 1 + 2 + ... + 10)*(1/2)^10
The sum of integers from 0 to 10 is (10 * 11) / 2 = 55/2
Expected number of boys = (55/2)*(1/2)^10
Expected number of boys = 55/2^11
To calculate the probability that the expected number of boys will not occur, you need to consider all the outcomes where the number of boys is different from the expected value. In this case, the expected number of boys is 55/2^11.
The number of outcomes where the number of boys is not equal to 5 (the expected value) can be calculated by subtracting the number of outcomes where the number of boys is equal to 5 from the total number of outcomes.
The total number of outcomes is 2^10 because each child can be either a boy or a girl, so there are two possibilities for each child.
The number of outcomes where the number of boys is 5 is calculated by selecting 5 boys out of the 10 children: C(10, 5) = 252.
Therefore, the probability that the expected number of boys will not occur is (2^10 - 252)/2^10.