Four persons are chosen at random from a group of 10 persons consisting of 4 men and 6
women. Three of the women are sisters. Calculate the probabilities that the four persons chosen
will
a) Consist of 4 women.
b) Consist of 2 women and 2 men
c) Include the three sisters
d) Do not include the three sisters.
a) prob(4 women)
= C(6,4)/C(10,4)
= 15/210
= 1/14
b) prob(2 women, 2 men)
= C(6,2)C(4,2)/C(10,4)
= 15(6)/210
= 3/7
c) with 3 sisters in, we need just one more of the remaining7 (could be man or woman)
prob(of event as stated)
= 1x C(7,1)/210
= 7/210 = 1/30
d)
prob(choosing any 4 of the remaining 7)
= C(7,4)/C(10,4)
= 35/210
= 1/6
Calculate the number of ways of selecting the four people if the are no restrictions
a) The probability of choosing 4 women from a group of 6 women is given by the combination formula:
P(4 women) = C(6, 4) / C(10, 4)
C(6, 4) represents the number of ways to choose 4 women from a group of 6, which is 15.
C(10, 4) represents the total number of ways to choose 4 people from a group of 10, which is 210.
P(4 women) = 15/210 = 1/14
So, the probability of choosing 4 women is 1/14.
b) The probability of choosing 2 women and 2 men can be calculated in a similar way:
P(2 women, 2 men) = (C(6, 2) * C(4, 2)) / C(10, 4)
C(6, 2) represents the number of ways to choose 2 women from a group of 6, which is 15.
C(4, 2) represents the number of ways to choose 2 men from a group of 4, which is 6.
C(10, 4) represents the total number of ways to choose 4 people from a group of 10, which is 210.
P(2 women, 2 men) = (15 * 6) / 210 = 90/210 = 3/7
So, the probability of choosing 2 women and 2 men is 3/7.
c) To calculate the probability of including the three sisters, we need to consider two cases: choosing exactly 3 women and choosing all 4 women.
P(including the three sisters) = P(3 women) + P(4 women)
P(3 women) = (C(6, 3) * C(4, 1)) / C(10, 4)
C(6, 3) represents the number of ways to choose 3 women from a group of 6, which is 20.
C(4, 1) represents the number of ways to choose 1 man from a group of 4, which is 4.
P(3 women) = (20 * 4) / 210 = 80/210 = 4/21
P(4 women) = 1/14 (calculated in part a)
P(including the three sisters) = 4/21 + 1/14 = 2/7
So, the probability of including the three sisters is 2/7.
d) The probability of not including the three sisters can be calculated by subtracting the probability of including the three sisters from 1:
P(not including the three sisters) = 1 - P(including the three sisters)
P(not including the three sisters) = 1 - 2/7 = 5/7
So, the probability of not including the three sisters is 5/7.
To calculate the probabilities for each scenario, we need to first determine the total number of possible outcomes and then calculate the number of favorable outcomes for each scenario.
a) Consist of 4 women:
Total number of possible outcomes = Choose(10, 4) = 10C4 = (10!)/(4!(10-4)!) = 210
Number of favorable outcomes = Choose(6, 4) = 6C4 = (6!)/(4!(6-4)!) = 15
Probability = Number of favorable outcomes / Total number of possible outcomes = 15/210 = 1/14
b) Consist of 2 women and 2 men:
Total number of possible outcomes = Choose(10, 4)
Number of favorable outcomes = Choose(6, 2) * Choose(4, 2) = (6!)/(2!(6-2)!) * (4!)/(2!(4-2)!) = 15 * 6 = 90
Probability = Number of favorable outcomes / Total number of possible outcomes
c) Include the three sisters:
To calculate the probability of including the three sisters, we need to consider the cases where we choose at least one sister.
Number of outcomes with at least one sister = Total number of outcomes - Number of outcomes without any sister
Total number of possible outcomes = Choose(10, 4)
Number of outcomes without any sister = Choose(7, 4) = (7!)/(4!(7-4)!) = 35
Number of outcomes with at least one sister = Total number of outcomes - Number of outcomes without any sister = Choose(10, 4) - Choose(7, 4)
Probability = Number of outcomes with at least one sister / Total number of possible outcomes
d) Do not include the three sisters:
To calculate the probability of not including the three sisters, we need to choose only from the remaining 3 women and 4 men.
Total number of possible outcomes = Choose(7, 4) * Choose(4, 0) = (7!)/(4!(7-4)!) * 1 = 35
Number of favorable outcomes = Choose(4, 4) * Choose(4, 0) = 1 * 1 = 1
Probability = Number of favorable outcomes / Total number of possible outcomes
Note: Choose(n, k) represents the number of ways to choose k items from a set of n items and is calculated as n!/(k!(n-k)!)