Probability

1. Probability (Yay!)

2. Probability lines • The probability of something happening is always expressed as a fraction or a decimal between 0 and 1, or a percentage. • The BIGGER the number, the more likely something is going to occur (guaranteed) • The smaller the #, the less likely it will occur (impossible) • Ex: 9/10 is way more likely than 1/100 • 0.75 is more likely than 0.25 • And 80% chance is better than a 30% chance.

3. Probability= desired outcome/ total # of outcomes • For a 6 sided di, there are 6 possibilities. And if we desire a 4. our prob looks like: • 1 (desired)/ 6 (possible)

4. Writing it down • How do we express a probability that something will occur? • P(event) • Ex 1: What’s the prob that you roll a 6? • It’s written like: P(6) = 1/6 • Ex 2: What’s the prob that you pick a heart? • P(H) = 13/52

5. Note • For this course, all games are fair, all decks start with 52 cards, there are no loaded dice (Oli), unless the question says differently!

6. Practice Questions • With a standard deck of 52 playing cards, find the probability that: • you will pick a Spade: • A 3 of spades: • A 9: • A red card: • *assume that you are replacing the card you pull out (this is known as “with replacement”)

7. What if… • What if we don’t replace the cards? • Prob that you pull out a 9: • Then another 9:

8. Multiplication rule • Used to determine the likelihood that two events will occur at the same time. • The words “And” or “Then” usually come up, • The Probability of A and B occurring = • P(A) X P(B) • Ex: The prob that A then B occurs= • P(A) X P(B)

9. Return to the previous problem • P(9): 4/52 • P(9’): 3/51 • So drawing two 9’s in a row looks like : P(9n9’): 4/52 X 3/51 = 12/2652 = 1/221

10. Questions • What’s the P of rolling a 5 then a 2? • What’s the probability that you draw 4 Kings in a row w/o replacement? • This means drawing a king, then a king, thena king, and then a king • What’s the prob that you will roll two 6’s in a row? • This means rolling a 6 then another 6

11. Questions • Prob of picking a 6 then a 7? • P(6)= 4/52 = 1/13 • P(7)= 4/51 • P(6 then 7)= 1/13 X 4/51 = 4/663

12. What’s the prob of drawing two Aces off the top of the deck? • P(A): 4/52 = 1/13 • P(A’): 3/51 = 1/17 • Prob of (A then A’)= 1/13 X 1/17 = 1/221

13. Remember • The probability of two things happening at the same time is the product of each of their probabilities. • P(A and B)= P(A) X P(B)

14. Addition Rule • Add the probabilities of two events if they can’t happen at the same time! • The words “Or” or “either” usually come up. • The Probability of A or B occurring = • P(A) + P(B) • The Probability of either of them occurring = P(A) + P(B)

15. Example • What’s the prob of Picking up a king or a queen from the deck of cards. • P(k) = 4/52 = 1/13 • P(q) = 1/13 • P(k or q)= 1/13 + 1/13 = 2/13

16. Oli’s Q • What’s the prob of choosing a heart or a spade? • P(H)= • P(S)= • P(H or S)= 13/52 + 13/52 • P(H + S)= 26/52= ½

17. Questions • What’s the prob of rolling a 1 or a 5 with one roll? • Prob of rolling a 3 or higher with one roll? • Prob of picking a red card or a spade? • Prob of picking either a Jack, Queen, or King? • Prob of rolling a 6 then an even number? • Prob of picking a Club then a 7 of hearts?

18. Word problems • Three friends are flipping two coins. Tina wins if both are heads. Gacia wins if both are tails. And Anais wins if there’s one of each. • a) is this game fair for all 3 players? • b) what are the probabilities of each?

19. Old Exam questions • The odds against an event occurring are 3 to 5. what is the probability that this event will not occur? • The odds against are 3:5 • The odds for are 5:3 • 5 success out of total 8 possibilities. • 5/8 • b) 3/5 • c) 3/8 • d) 5/8

20. 4:6 odds

21. 3:7, 5:2, 1:9 • The first number indicates a success. • 3:7= 3/10 • 5:2= 5/7 • 1:9= 1/10

22. Exam q’s • A school held a draw to award a snowboard. Marie had 3 tickets, Christophe had 4 tickets and Olympia had 7 tickets. • What are the odds that the wining ticket was Marie’s? • a) 1 to 2 c) 3 to 11 • b) 1 to 3 d) 3 to 14

23. Team Ramrod has 4 tickets, team brandons has 12 tickets, and team america has 5 tickets. What are the odds that team brandon will win? 12:9 • odds that he wont win? 9:12 • Probability that he wins? • 12:9. how many wins? 12 wins possible • Out of how many total possibilities? 21 • 12/21

24. Maia decided to buy a pack of 100 CD’s. However, 10 of them were defective. • Maia chose two CD’s at random. • Given that both of the CD’s were in the same condition, what is the probability that they were defective? • a) 1/11 c) 1/110 • b) 1/90 d) 89/110

25. Show all your work • A ski resort offers 2 types of passes: weekday pass and unlimited pass. This year they sold 450 passes. • If one of the resort’s pass holders is chosen at random, the prob that it will be a man is 3/5. • If a female is chosen at random, the prob that she will have a weekday pass is 40%. • If a male pass holder is chosen at random, the prob the he will have an unlimited pass is 5/9. • How many weekday passes were sold?

26. A pet shop has 60 mice for sale. Some are female, and some are male. These mice are either white or grey. • If a mouse is chosen at random, the prob that it will be white is 3/5. • If a grey mouse is chosen at random, the prob that it will be a female is 0.25. • The # of white female mice is twice the number of grey female mice. • Victor randomly chose a mouse at this pet shop. • What’s the probability that he chooses a female mouse?

27. James and Dan are taking their students white water rafting. • There will be 9 students in James’ boat and 11 in Dan’s. There will be more boys in James’ boat than in Dan’s. • Two students chosen at random will each win a free excursion ticket. One will be chosen from each boat. • James has determined that the prob that the 2 free tickets will be won by 2 boys is 35/99. • what’s the prob that the free tix will be won by 2 girls?