a If the order in which people are chosen is not important, we should use combinations.
B
b If the order in which people are chosen is important, we should use permutations.
A
a 1/220 or ≈ 0.005
B
b 1/1320 or ≈ 0.0008
Practice makes perfect
a Three volunteers are chosen at random from a group of 12 to help at a summer camp. In order to find the probability that you, your brother, and your friend are chosen, we will use the theoretical probability. We will compare the number of favorable outcomes to the number of possible outcomes.
P = favorable outcomes/possible outcomes
The number of possible outcomes can be found using the formula for the number of combinations of 12 people chosen 3 at a time.
_(12)C_3
We use combinations because the order in which the 3 people are chosen is not important. Let's recall the formula to calculate _nC_k.
_nC_k =n!/k!(n-k)!
We can substitute 12 for n and 3 for k.
Now we need to find the number of favorable outcomes. Since only one of the possible combinations is you, your brother, and your friend, the number of favorable outcomes is 1. We are ready to find the desired probability.
The probability that you, your brother, and your friend are chosen at random from a group of 12 is 1220, or about 0.005.
b Once again, we are choosing 3 volunteers at random from a group of 12. As in Part A, we will need to find the theoretical probability of the given situation.
P = favorable outcomes/possible outcomes
This time the order of the people chosen is important — the first is a counselor, the second is a lifeguard, and the third is a cook. We are interested in finding the probability that you are a cook, your brother is a lifeguard, and your friend is a counselor. The number of possible outcomes will be the permutations of 3 people chosen from a group of 12.
_(12)P_3
We will use permutations because the order is important. Let's recall the formula to calculate _nP_k.
_nP_k =n!/(n-k)!
We can substitute 12 for n and 3 for k and simplify.
Since there is only possible permutation where you are the cook, your brother is the lifeguard, and your friend is the counselor, the number of favorable outcomes is 1. We are ready to find the desired probability.
