A druggist wishes to select three brands of aspirin to sell in his store. He has five major brands to choose from: A, B, C, D and E. If he selects the three brands at random, what is the probability that he will select the following?
(a) brand B
(b) brands B and C
(c) at least one of the two brands B and C
To find the probability of selecting certain combinations of brands, we need to determine the total number of possible outcomes and the number of favorable outcomes for each case.
There are a total of 5 brands to choose from, and the druggist wishes to select 3 brands. Therefore, the total number of possible outcomes or combinations is given by the binomial coefficient:
C(5, 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10.
Now let's calculate the probabilities for each case:
(a) Probability of selecting brand B:
There is only one brand B, and since we are selecting 3 brands in total, the number of favorable outcomes is 1.
Therefore, the probability of selecting brand B is 1/10.
(b) Probability of selecting brands B and C:
We need to choose both brands B and C, which means we have two specific brands to select. The number of favorable outcomes in this case is 1 (since we are selecting specific brands).
Therefore, the probability of selecting brands B and C is 1/10.
(c) Probability of selecting at least one of the two brands B and C:
To calculate this, we need to find the probability of not selecting both B and C.
The complement of not selecting both B and C is selecting at least one of B and C.
The number of favorable outcomes in this case would be the total number of outcomes minus the outcomes of not selecting both B and C.
Not selecting both B and C means selecting from the remaining 3 brands (A, D, and E).
The number of combinations when selecting from 3 brands is C(3, 3) = 1.
Therefore, the number of favorable outcomes for selecting at least one of B and C is 10 - 1 = 9.
Hence, the probability of selecting at least one of the two brands B and C is 9/10.
To summarize:
(a) Probability of brand B: 1/10
(b) Probability of brands B and C: 1/10
(c) Probability of at least one of B and C: 9/10