Assuming that the probability of male birth is 0.5,find the probability that in the family of 4 children there will be
A. 1 boy
B. 2 boys
C. At least a boy
0.125
To find the probability of certain outcomes in a family of 4 children, we can use the concept of binomial probability.
A. Probability of having 1 boy:
The probability of having a boy in any given birth is 0.5, and since we have 4 children, we can use the binomial probability formula. The formula for the probability of exactly k successes in n trials is:
P(k) = (nCk) * p^k * (1 - p)^(n - k)
Where nCk is the binomial coefficient, p is the probability of success, and (1 - p) is the probability of failure.
In this case, we want to find the probability of having 1 boy (k = 1) out of 4 children (n = 4), with the probability of success (having a boy) being 0.5 (p = 0.5).
P(1) = (4C1) * (0.5)^1 * (1 - 0.5)^(4 - 1)
Now, let's calculate P(1):
P(1) = (4C1) * (0.5)^1 * (0.5)^(3)
= 4 * 0.5 * 0.125
= 0.25
Therefore, the probability of having exactly 1 boy in a family of 4 children is 0.25.
B. Probability of having 2 boys:
Similarly, using the binomial probability formula, we can calculate the probability of having exactly 2 boys (k = 2) out of 4 children (n = 4) with a probability of success (having a boy) being 0.5 (p = 0.5).
P(2) = (4C2) * (0.5)^2 * (1 - 0.5)^(4 - 2)
Calculating P(2):
P(2) = (4C2) * (0.5)^2 * (0.5)^(2)
= 6 * 0.25 * 0.25
= 0.375
Therefore, the probability of having exactly 2 boys in a family of 4 children is 0.375.
C. Probability of having at least 1 boy:
In this case, we want to find the probability of having at least 1 boy in a family of 4 children. To calculate this, we can find the probability of having no boys and subtract it from 1.
Probability of having no boys (all girls):
P(no boys) = (0.5)^4
P(at least 1 boy) = 1 - P(no boys)
= 1 - (0.5)^4
= 1 - 0.0625
= 0.9375
Therefore, the probability of having at least 1 boy in a family of 4 children is 0.9375.
To calculate the probability of different outcomes in the family of 4 children, we can use the concept of binomial probability.
A. Probability of having 1 boy:
In this case, there are 4 children and we want exactly 1 of them to be a boy. The probability of having a boy is 0.5. So, we can use the binomial probability formula:
P(exactly 1 boy) = (number of ways to have 1 boy) * (probability of having a boy) * (probability of having girls for the remaining children)
Number of ways to have 1 boy out of 4 children can be calculated using combination formula or binomial coefficient. The formula to calculate the combination is:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of children and r is the number of boys we want.
C(4, 1) = 4! / (1! * (4 - 1)!) = 4
Now, we can calculate the probability:
P(exactly 1 boy) = (number of ways to have 1 boy) * (probability of having a boy) * (probability of having girls for the remaining children)
= 4 * 0.5 * 0.5 * 0.5 * 0.5
= 0.25
So, the probability of having exactly 1 boy in a family of 4 children is 0.25.
B. Probability of having 2 boys:
Similarly, we can calculate the probability of having 2 boys using the same formula:
P(exactly 2 boys) = (number of ways to have 2 boys) * (probability of having a boy) * (probability of having girls for the remaining children)
Number of ways to have 2 boys out of 4 children can be calculated as:
C(4, 2) = 4! / (2! * (4 - 2)!) = 6
Now, we can calculate the probability:
P(exactly 2 boys) = 6 * 0.5 * 0.5 * 0.5 * 0.5
= 0.375
So, the probability of having exactly 2 boys in a family of 4 children is 0.375.
C. Probability of having at least a boy:
To find the probability of having at least a boy in a family of 4 children, we can calculate the probability of having no boys and subtract it from 1.
P(at least a boy) = 1 - P(no boys)
With no boys, the possible outcome is having all girls, which can be calculated as:
P(no boys) = (probability of having a girl) * (probability of having a girl) * (probability of having a girl) * (probability of having a girl)
= 0.5 * 0.5 * 0.5 * 0.5
= 0.0625
Therefore,
P(at least a boy) = 1 - 0.0625
= 0.9375
So, the probability of having at least a boy in a family of 4 children is 0.9375.