It is assumed that when children are born they are equally likely to be boys or girls.What is the probability that a family of four children contains (a) three boys and one girl (b) two boys and two girls
both are binomial distribution problems
prob(boy) = 1/2
prob(girl)
a) in 4 children, we want 3 boys and 1 girl
= C(4,3)(1/2)^3 (1/2)^1
= 4(1/16) = 1/4
b)2 boys, 2 girls
= C(4,2)(1/2)^2 (1/2)^2
= 6(1/16) = 3/8
To determine the probability of a family of four children having a particular combination of boys and girls, we can use the concept of binomial probability.
The probability of having a boy or a girl is equal and is 1/2 since it is assumed that children are equally likely to be boys or girls.
(a) To calculate the probability of having three boys and one girl in a family of four children, we need to consider all possible ways this combination can occur. There are four ways this can happen: BBBG, BBGB, BGBB, and GBBB. Each of these possibilities has an equal probability of occurring.
The probability of having three boys and one girl is calculated by multiplying the probabilities for each of the possible combinations:
P(3 boys and 1 girl) = P(BBBG) + P(BBGB) + P(BGBB) + P(GBBB)
Since the probability of having a boy or girl is 1/2, the probability for each of the combinations is (1/2)^4 = 1/16.
P(3 boys and 1 girl) = 1/16 + 1/16 + 1/16 + 1/16 = 4/16 = 1/4
Therefore, the probability of a family of four children having three boys and one girl is 1/4.
(b) Similarly, to calculate the probability of having two boys and two girls in a family of four children, we need to consider all possible ways this combination can occur. There are six ways this can happen: BBGG, BGBG, BGGB, GBBG, GBGB, and GGBB. Each of these possibilities has an equal probability of occurring.
Using the same reasoning as before, the probability of having two boys and two girls is:
P(2 boys and 2 girls) = P(BBGG) + P(BGBG) + P(BGGB) + P(GBBG) + P(GBGB) + P(GGBB)
The probability for each of the combinations is (1/2)^4 = 1/16.
P(2 boys and 2 girls) = 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 = 6/16 = 3/8
Therefore, the probability of a family of four children having two boys and two girls is 3/8.
To calculate the probability of different combinations of boys and girls in a family of four children, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
(a) Three boys and one girl:
To calculate the probability of having three boys and one girl, we need to determine the number of favorable outcomes.
Number of favorable outcomes:
There are four positions to be filled with either a boy (B) or a girl (G). We can choose one of the positions for a girl in 4C1 ways and the remaining three positions for boys in 3C3 ways.
Number of favorable outcomes = 4C1 * 3C3 = 4 * 1 = 4
Total number of possible outcomes:
For each child, there are two possibilities: boy (B) or girl (G). Since each child's gender is independent of the others, we can multiply the number of possibilities for each child: 2 * 2 * 2 * 2 = 2^4 = 16
Therefore, the probability of having three boys and one girl is:
Probability = Number of favorable outcomes / Total number of possible outcomes = 4 / 16 = 1 / 4 = 0.25 (or 25%)
(b) Two boys and two girls:
To calculate the probability of having two boys and two girls, we again need to determine the number of favorable outcomes.
Number of favorable outcomes:
There are four positions to be filled with either a boy (B) or a girl (G). We can choose two of the positions for boys in 4C2 ways, and the remaining two positions for girls in 2C2 ways.
Number of favorable outcomes = 4C2 * 2C2 = 6 * 1 = 6
Total number of possible outcomes:
Similarly, for each child, there are two possibilities: boy (B) or girl (G). We multiply the number of possibilities for each child to get the total number of possible outcomes: 2 * 2 * 2 * 2 = 2^4 = 16
The probability of having two boys and two girls is:
Probability = Number of favorable outcomes / Total number of possible outcomes = 6 / 16 = 3 / 8 = 0.375 (or 37.5%)