What is Probability?

1. What is Probability? • The study of probability helps us figure out the likelihood of something happening. • In math we call this “something happening” or an “event” • Probability is a way of expressing what the chances are that an event will occur

2. The term probability is common • Weather forecast - 30 % chance of rain • Gambling - two dice will produce a sum of 7 is 1/6 Things to know: 1st thing: What the probability statements mean 2nd thing: Then we consider how we determine the numerical value for that probability

3. Example: Number of ways 2 dice would produce a sum of 7

4. 36 different combinations • Shows all the different ways the dice can land • Shows all the ways you could roll a seven • 6 of the 36 produce the sum of 7 • Must assume that each of the 36 combinations are equally likely to occur • So there is a 1/6 chance that you the sum of the dice will be 7

5. Example: Jar with 4 red marbles and 6 blue marbles Want to find the probability of drawing a red marble at random. Favorable outcome = drawing a red marble

6. Ways to Express Probability As a fraction~ 4/10 = 2/5 As a decimal ~ 4/10 = .4 As a percent ~ 4/10 = 40/100 = 40% Unlikely events have a probability near zero Likely events have probabilities near 1

7. What is the total number of possible outcomes? • Is called a sample space • Sample space is a set consisting of all the possible outcomes of an event. The number of different ways you can choose something from the sample space is the total number of possible outcomes.

8. We only have 2 events with our red and blue marbles • Either we pick a red marble or a blue marble • If you don’t do the first, then you must do the second • So the probability of picking a red marble plus the probability of picking a blue marble equals 1 or 100% Sum = 4/10 + 6/10 = 10/10 = 1

9. So we have 4 red marbles and 6 blue marbles in our jar Sample space = all ten marbles because we are likely to draw any one of them Favorable outcomes = # of red marbles = 4 Possible outcomes = total # of marbles = 10 4/10 reduces to 2/5 Probability of drawing a red marble where all outcomes are equally likely is 2/5 (sample space)

10. When one event occurs: • Probability of picking a red marble was 4/10 or 2/5 • Sample space = 10 marbles in the jar • So the probability of not picking a red marble 1 = 10/10 10/10 -4/10 = 6/10 or 3/5 (this is also the probability of picking a blue marble)

11. When two events that are equally likely occur: • You draw 1 marble from the 10 • Then I draw another marble from the nine that remain What is the probability that I will draw a blue one first? What is the probability that you will draw a red one second?

12. Your probability of drawing a blue one is 6/10 • After you draw there are only 9 marbles left and 4 of those are red, so the probability that I will draw a red one is 4/9 When there are 2 events, the second outcome is dependent on the first.

13. When two or more events occur that are not all equally likely: A. you draw a blue marble and then I draw a blue marble B. you draw a blue marble and then I draw a red marble C. you draw a red marble and then I draw a blue marble D. you draw a red marble and then I draw a red marble There are four possibilities but they are not all equally likely. Two separate events with the work “and”, there the outcome of the second is dependent on the outcome of the first we multiply.

14. Example A: Your probability of drawing a blue marble (6/10 = 3/5) X my probability of drawing another blue marble which would be 5/9

15. Example B: Your probability of drawing a blue marble (3/5) X my probability of drawing a red marble (4/9)

16. Classroom Exercise What can you learn from the chart?

17. Average Distribution of Colors