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TRANSCRIPT
Using Counting Techniques
` In calculating probabilities, P(A), we usually: ◦ List the outcome set, S; ◦ List the event set, A; and ◦ Count the number of elements in each list, that is,
n(S) and n(A).
` There are useful techniques to aid us in accomplishing this: ◦ Tree diagrams ◦ Permutations ◦ Combinations
n(S)n(A)P(A)
1. A box contains 3 defective light bulbs and 7 good bulbs. a) Construct a tree diagram illustrating the various
possibilities relating to 3 consecutive bulbs being drawn at random from the box without replacement.
b) Find the probability that at least one good bulb is drawn from the box.
P(at least 1 good bulb) = P(exactly 3 good) + P(exactly 2 good) + P(exactly 1 good)
P(exactly 3 good) = P(GGG) = 0.29 P(exactly 2 good) = P(GGB) + P(GBG) + P(BGG) = 0.525 P(exactly 1 good) = P(GBB) + P(BGB) + P(BBG) = 0.175 P(at least 1 good bulb) = 0.29 + 0.525 + 0.175 = 0.99
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It would be easier to calculate P(at least 1 good bulb) using an indirect method:
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` Tree diagrams often become quite cumbersome and complicated.
` Therefore, we may bypass listing the outcome and event sets and just calculate the size of the sets directly, that is, n(S) and n(A), by using counting principles based on permutations and combinations.
2. Nine horses are entered in a race. In an attempt to predict the finish of the race, three horses are selected by lot to finish first, second, and third. What is the probability that the choice is correct?
` The outcome set consists of the ways that three horses chosen from nine can place first, second, and third.
` The event set contains exactly one element, the outcome where the order is correct.
504Pn(S) 39
1n(A)
.504
1iscorrectisfinishtheofchoicethethatyprobabilitthe
5041P(A)
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3. A committee of five medical professionals is to be selected from ten doctors and eight nurses. What is the probability that there are exactly three doctors on the committee? (Assume each outcome is equally likely.)
` The outcome set is to select 5 from a set of 18 (10 doctors, 8 nurses).
` The event set is to select 3 doctors from 10 doctors that leaves 2 nurses from 8 nurses.
` The probability of selecting a committee with exactly three doctors is approximately 0.4 or 40%.
8568Cn(s) 518
336028120CCn(A) 28310
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0.485683360P(A)
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