You are 1 of 7 female candidates auditioning for 2 parts-the...

You are 1 of 7 female candidates auditioning for 2 parts-the heroine and her best friend-in a play. Before the auditions you know nothing of the other candidates, and you assume all candidates have equal chances for the parts. a. How many distinct choices are possible for casting the two parts? b. In how many of the possibilities in part (a) would you be chosen to play the heroine? c. In how many of the possibilities in part (a) would you be chosen to play the best friend? d. Use the results in parts (a) and (b) to find the probability that you will be chosen to play the heroine. Indicate a more direct way of finding this probability. e. Use the results in parts (a), (b), and (c) to find the probability that you will be chosen to play 1 of the 2 parts. Indicate a more direct way of finding this probability.

You are 1 of 7 female candidates auditioning for 2 parts-the heroine and her best friend-in a play. Before the auditions you know nothing of the other candidates, and you assume all candidates have equal chances for the parts.
a. How many distinct choices are possible for casting the two parts?
b. In how many of the possibilities in part (a) would you be chosen to play the heroine?
c. In how many of the possibilities in part (a) would you be chosen to play the best friend?
d. Use the results in parts (a) and (b) to find the probability that you will be chosen to play the heroine. Indicate a more direct way of finding this probability.
e. Use the results in parts (a), (b), and (c) to find the probability that you will be chosen to play 1 of the 2 parts. Indicate a more direct way of finding this probability.

Key Concepts

Sample Space

The sample space in a probability problem represents the set of all possible outcomes. Here, it consists of every distinct way the two roles can be assigned to the candidates. Accurately defining the sample space is crucial, as it serves as the denominator in probability calculations and ensures that comparisons of favorable and total outcomes are valid.

Probability Calculation

Probability calculation is about determining the likelihood of an event by comparing the number of favorable outcomes to the total number of possible outcomes. In casting scenarios, once the total number of arrangements (the sample space) is determined using counting methods, specific events (such as being cast in a given role) can be assessed by counting the number of outcomes that satisfy the event's conditions and dividing by the overall count.

Counting Outcomes

Counting outcomes involves determining the total number of distinct possibilities based on specific rules or criteria. In such audition scenarios, one must account for the use of permutation techniques when roles are distinct, ensuring that no potential arrangement is double?counted or overlooked. This systematic approach to counting is foundational in combinatorial problems.

Permutation

Permutations refer to the arrangements of a set of objects in a specific order. In the context of this problem, since the roles of heroine and best friend are distinct, the order in which the candidates are chosen matters. This means that even if the same two candidates are involved, assigning one as heroine and the other as best friend is considered a different permutation from swapping their roles.

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