8. A survey was conducted to learn about the shopping preference of consumers during the

holiday season. Out of 2000 respondents 22% said they would prefer to shop online
because of the free shipping offered; 45% said that they would like to shop in malls and
discount outlets while 15% of the shoppers said they would use both. The probability
that a randomly selected shopper would use at least one method of shopping is:
(a) 67%
(b) 52%
(c) 82%
(d) 60%
(e) 80%
9. In the Powerball lottery, a player first selects five different numbers between 1 and 59.
After selecting the five numbers, the player then selects one number between 1 and 35.
This number is known as the Powerball. To calculate the odds of winning, we multiply the
number of ways the five numbers can be drawn from 1 and 59 by 35. The number of
ways the winning Powerball number can be drawn or the chance of winning the
Powerball is 1 divided by one of the following numbers:
(a) 5,006,386
(b) 230,000,000
(c) 175,223,510
(d) 180,345,876
(e) 190,678,000

8. To find the probability that a randomly selected shopper would use at least one method of shopping, we need to calculate the complement of the probability that a randomly selected shopper would not use any method of shopping.

The percentage of shoppers who prefer to shop online because of free shipping is 22%. The percentage of shoppers who prefer to shop in malls and discount outlets is 45%. The percentage of shoppers who would use both methods is 15%.

Therefore, the percentage of shoppers who would not use any method of shopping is:

100% - (22% + 45% - 15%) = 100% - 62% = 38%

The probability that a randomly selected shopper would not use any method of shopping is 38%.

Therefore, the probability that a randomly selected shopper would use at least one method of shopping is:

100% - 38% = 62%

So the answer is (d) 62%.

9. To calculate the odds of winning the Powerball, we need to multiply the number of ways the five numbers can be drawn from 1 to 59 (which is 59 choose 5) by 35 (the number of ways the Powerball number can be drawn from 1 to 35).

Using combination formula, 59 choose 5 can be calculated as:

59! / (5! * (59-5)!) = (59 * 58 * 57 * 56 * 55) / (5 * 4 * 3 * 2 * 1) = 5,006,386

So the number of ways the winning Powerball number can be drawn is 5,006,386.

The chance of winning the Powerball is the reciprocal of this number, so:

1 / 5,006,386

Therefore, the answer is (a) 5,006,386.

8. To find the probability that a randomly selected shopper would use at least one method of shopping, we need to calculate the complement of the probability that a shopper would not use either method.

Let's break down the information given:
- 22% of the shoppers prefer to shop online because of the free shipping.
- 45% prefer to shop in malls and discount outlets.
- 15% of the shoppers use both methods.

To find the percentage of shoppers who do not use either method, we subtract the percentages of those who use each method along with those who use both methods from 100%.

Percentage of shoppers who do not use either method = 100% - (22% + 45% - 15%)
= 100% - (67%)
= 33%

Therefore, the probability that a randomly selected shopper would use at least one method of shopping is 100% - 33% = 67%.

The correct option is (a) 67%.

9. In the Powerball lottery, a player selects five different numbers between 1 and 59, and then selects one number between 1 and 35, known as the Powerball. To calculate the odds of winning, we need to find the total number of possible outcomes and the number of desired outcomes.

The total number of possible outcomes when selecting five numbers from 1 to 59 is the combination of 59 numbers taken 5 at a time. This can be calculated using the formula:

Total number of outcomes = C(59, 5) = 59! / (5! * (59-5)!)

The total number of possible outcomes when selecting one number from 1 to 35 for the Powerball is 35.

To calculate the odds of winning the Powerball, we divide the number of desired outcomes (1) by the total number of possible outcomes.

Therefore, the chance of winning the Powerball is 1 divided by the total number of outcomes, which is:

1 / (C(59, 5) * 35)

Calculating the exact value of this expression would be time-consuming. However, we can identify the option that matches the number obtained.

The correct option is:

(d) 180,345,876

Therefore, the chance of winning the Powerball is 1 divided by 180,345,876.