## Introduction to Probability

First, we will discuss experiments where outcomes are equally likely to occur and the frequency approach to assigning probabilities. Then, we will focus on the concept of events and touch on the issue of conditional probability.

1. 3 out of the 15 possible outcomes are favorable, so 3/15 = 0.2

2. There are 7+3 = 10 favorable outcomes out of a possible 15. 10/15 = 0.667

3. The probability of drawing a red marble is 3/15 = 0.2, so the probability of NOT drawing a red marble is: 1 - 0.2 = 0.8.

4. To solve this problem, list all the possible outcomes. There are 36 of them since each die can come up one of six ways. There are four ways to get a sum of five: (1) Get a 1 then a 4 (2) Get a 4 then a 1 (3) Get a 2 then a 3 (4) Get a 3 then a 2; 4/36 = 0.11

5. Two events are independent if the occurrence of one has no effect on the probability of the occurrence of the other. A student's test grades are not independent because a student who does well on the midterm understands the material and is therefore more likely to do well on the final. Two draws of an ace without replacement are not independent because what you get on your first draw affects the probability of getting an ace on the second draw. This is a conditional probability.

6. P(Odd # AND Tails) = 3/6 x 1/2 = .5 x .5 = 0.25

7. The probability that you win $1 OR$3 (or both games) is: (1/2) + (1/6) - (1/2)(1/6) = 7/12 = 0.58

8. The probability that person 1 AND person 2 AND person 3 will take public transportation is: .6 x .6 x.6 = 0.216

9. The probability of getting a 3 on the first roll OR a 3 on the second roll is: (1/6) + (1/6) - (1/6)(1/6) = 11/36 = 0.31

10. The probability of NOT getting a 4 on any roll is: (5/6)(5/6)(5/6)(5/6) = .48, so the probability of rolling at least one 4 is: 1 - .48 = 0.52

11. P(spade on the first draw) = 13/52 = .25, P(spade on the second draw | spade on the first draw) = 12/51 = .235, .25*.235 = 0.0588

12. They are both equally likely. There are 8 possible outcomes for flipping a coin 3 times, so each has a 1/8 probability.