Exercise 48 Page 705

We want to find the probability of winning a lottery. To do so we will use theoretical probability. P=Favorable Outcomes/Possible Outcomes We will start by finding the number of possible outcomes. The lottery numbers are selected at random and the order is important. We are choosing among integers from 0 to 9. 0 1 2 3 4 5 6 7 8 9 Therefore, there are 10 numbers to choose from. We are going to start by finding the number of possible outcomes. It will be the number of permutations of 4 integers taken from the total of 10 integers because the order is important. Let's recall the formula for the number of permutations of n objects taken r at a time, where r ≤ n. _n P_r = n!/(n-r)! In our case, the total number of integers in the lottery is 10, so n = 10. Out of them we choose 4 numbers, therefore r = 4. Let's substitute these values into the formula.

_nP_r = n!/(n-r)!

SubstituteII

n= 10, r= 4

_(10)P_4 = 10!/( 10- 4)!

▼

Evaluate right-hand side

SubTerm

Subtract term

_(10)P_4 = 10!/6!

Write as a product

_(10)P_4 = 10 * 9 * 8 * 7 * 6!/6!

CancelCommonFac

Cancel out common factors

_(10)P_4 = 10 * 9 * 8 * 7 * 6!/6!

SimpQuot

Simplify quotient

_(10)P_4 = 10 * 9 * 8 * 7/1

DivByOne

a/1=a

_(10)P_4 = 10 * 9 * 8 * 7

Multiply

Multiply

_(10)P_4 = 5040

Next, we will look for the number of favorable outcomes. There is only one permutation of numbers that guarantees winning in the lottery, so the number of favorable outcomes is 1. This means that we have enough information to calculate the desired probability. P=Favorable Outcomes/Possible Outcomes [0.6em] ⇕ P=1/5040 The probability of winning the lottery is 15040.