1. A bag contains 12 red checkers and 6 black checkers. You will randomly select two checkers, one at a time, without replacement.
(a) In how many ways can you select two checkers so that at least one of the checkers is red? List the ways.
(b) What is the probability that you select a black checker and then a red checker? Show your work.
To find the number of ways to select two checkers so that at least one of them is red, we can use the concept of combinations.
(a) To calculate the number of ways to select two checkers, we'll consider two scenarios: selecting two red checkers, and selecting one red and one black checker.
1. Selecting two red checkers:
Since there are 12 red checkers, the number of ways to select two red checkers is given by the combination formula: C(12, 2) = 12! / (2! * (12-2)!) = 66 ways.
2. Selecting one red and one black checker:
We have 12 red checkers and 6 black checkers. To find the number of ways to select one red checker and one black checker, we multiply the number of choices for each color. Thus, the number of ways to select one red and one black checker is 12 * 6 = 72 ways.
Therefore, the total number of ways to select two checkers so that at least one of them is red is the sum of the two scenarios: 66 + 72 = 138 ways.
The ways to select two checkers so that at least one of them is red are as follows:
1. Selecting two red checkers (66 ways)
2. Selecting one red checker and one black checker (72 ways)
(b) To find the probability of selecting a black checker and then a red checker, we need to calculate the favorable outcomes and divide by the total number of possible outcomes.
1. Favorable outcomes:
The probability of selecting a black checker first is 6/18 (since there are 6 black checkers out of a total of 18 checkers after one is already chosen). After selecting a black checker, there are 12 red checkers remaining out of a total of 17 checkers. So, the probability of selecting a red checker second is 12/17.
Multiplying these probabilities together gives: (6/18) * (12/17) = 2/17.
2. Total possible outcomes:
The total number of ways to select two checkers is given by C(18, 2) = 18! / (2! * (18-2)!) = 153.
Therefore, the probability of selecting a black checker and then a red checker is 2/17, or approximately 0.1176 (rounded to four decimal places).
(a) 6 combinations of Red/Black and 3 combinations of Red/Red
(b) There is a 6/18 chance for a Black Checker being drawn so 1/3.
There is a 12/18 chance for a Red Checker being drawn so 2/3.
(If you draw a black first then the chance of drawing a red drops to 12/17 because a black checker is removed from play.)