Exercise 45 Page 705

We want to find the probability that you and your friend are chosen from an audience on a television game show. To do so, we will use the theoretical probability. P=Favorable Outcomes/Possible Outcomes We start by finding the number of possible outcomes. This will be the number of combinations in which we can choose 2 contestants from an audience of 300 people. The order in which we choose them is not important, since we consider them as a pair of equally important contestants. Let's recall the formula for the number of combinations of n objects taken r at a time. _nC_r = n!/(n-r)! * r! In our case there are 300 people in the audience, so n= 300. Out of them, 2 people are randomly selected as contestants. Therefore, we know that r = 2. Let's substitute these values and find the number of possible combinations.

_nC_r = n!/(n-r)! * r!

SubstituteII

n= 300, r= 2

_(300) C_2 = 300!/( 300- 2)! * 2!

▼

Evaluate right-hand side

SubTerm

Subtract term

_(300)C_2 = 300!/298! * 2!

Write as a product

_(300)C_2 = 300 * 299 * 298!/298! * 2!

CancelCommonFac

Cancel out common factors

_(300)C_2 = 300 * 299 * 298!/298! * 2!

SimpQuot

Simplify quotient

_(300)C_2 = 300 * 299/2!

2!=2

_(300)C_2 = 300 * 299/2

Multiply

Multiply

_(300)C_2 = 89 700/2

CalcQuot

Calculate quotient

_(300)C_2 = 44 850

Therefore, the number of possible outcomes is 44 850. Next, we will look for the number of favorable outcomes. Notice there is only one combination that chooses you and your friend as contestants, so the number of favorable outcomes is 1. We have enough information to calculate the desired probability. P=Favorable Outcomes/Possible Outcomes [0.9em] ⇕ P=1/44 850 The probability that you and your friend are chosen as contestants is 144 850.