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. Probability Problems • Problem: A spinner has 4 equal sectors colored yellow, blue, green, and red. What are the chances of landing on yellow after spinning the spinner? Solution: The chances of landing on yellow are 1 in 4, or one fourth.

4. Probability Problems • Very important to identify all of the different outcomes that could occur • Outcomes = the different results that could happen in the event or the different possibilities • To find a basic probability with all outcomes equally likely, we use a fraction: number of favorable outcomes total number of possible outcomes Probability =

5. What is a favorable outcome? • Example: A spinner has 4 equal sectors colored yellow, blue, green and red.Want to find the probability of the spinner landing on yellow. • Favorable outcome =landing on yellow section The probability of landing on yellow is 1 in 4 or 1/4

6. Experiment: A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color? Possible Outcomes: The possible outcomes of this experiment are yellow, blue, green, and red. Probabilities: P(yellow)  =  number of ways to land on yellow  = 1 total number of colors 4    P(blue)  =  number of ways to land on blue  =  1 total number of colors   4    P(green)  =  number of ways to land on green  =  1 total number of colors   4    P(red)  =  number of ways to land on red  =  1 total number of colors 4 

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

8. 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

9. 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.

10. 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

11. 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)

12. 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

13. Probability is a fraction of the sample space • The sum of the probabilities of all the possible outcomes equals 1 • The probability of the occurrence of an event is always 1 minus the probability that it doesn’t occur

14. P(yellow)  =  number of ways to land on yellow  = 1 total number of colors 4    P(blue)  =  number of ways to land on blue  =  1 total number of colors   4    P(green)  =  number of ways to land on green  =  1 total number of colors   4    P(red)  =  number of ways to land on red  =  1 total number of colors 4  • Probability = ¼ • Sample space = 4 colors • Sum = ¼ + ¼ + ¼ + ¼ = 1 • Probability of NOT landing on yellow = 1-1/4 = 3/4 Spinner problem

15. Marble Problem • 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)

16. 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

17. Situation with 2 events • 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?

18. 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.

19. Classroom Exercise/Homework What can you learn from the chart?

20. %’s of colors in different kinds of M & M’s 1/2007