The document defines probability and provides examples of calculating probabilities using tree diagrams and tables. It discusses key probability concepts like: - Probability is defined as the number of desired outcomes divided by the total number of possible outcomes and must be between 0 and 1. - Tree diagrams can show all possible outcomes of multiple events and be used to calculate joint probabilities. - Independent events do not influence each other, so the probability of one event does not depend on whether the other occurs. - Examples include coin flips, rolling dice, and drawing colored beads from a bag to demonstrate calculating probabilities of single and multiple events.

2. Probability definition The probability of an event occurring is: Where E is the set of desired events (outcomes) Where S is the set of all possible events (outcomes) Note that 0 ≤ |E| ≤ |S| Thus, the probability will always between 0 and 1 An event that will never happen has probability 0 An event that will always happen has probability 1 2

3. What are Tree Diagrams A way of showing the possibilities of two or more events Simple diagram we use to calculate the probabilities of two or more events

4. For example – a fair coin is spun twice Possible Outcomes 2nd 1st H HH H T HT H TH T T TT

5. Probability is always a value between 0 and 1 Something with a probability of 0 will never occur Something with a probability of 1 will always occur You cannot have a probability outside this range! Note that when somebody says it has a “100% probability) That means it has a probability of 1 5

6. Attach probabilities 2nd 1st H HH P(H,H)=½x½=¼ ½ H ½ ½ T HT P(H,T)=½x½=¼ H TH ½ P(T,H)=½x½=¼ ½ T ½ T TT P(T,T)=½x½=¼ INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin

7. Calculate probabilities 2nd 1st * H HH P(H,H)=½x½=¼ ½ H ½ ½ * T HT P(H,T)=½x½=¼ * H TH ½ P(T,H)=½x½=¼ ½ T ½ T TT P(T,T)=½x½=¼ Probability of at least one Head? 3/4

8. Dice probability What is the probability of getting “snake-eyes” (two 1’s) on two six-sided dice? Probability of getting a 1 on a 6-sided die is 1/6 Via product rule, probability of getting two 1’s is the probability of getting a 1 AND the probability of getting a second 1 Thus, it’s 1/6 * 1/6 = 1/36 What is the probability of getting a 7 by rolling two dice? There are six combinations that can yield 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Thus, |E| = 6, |S| = 36, P(E) = 6/36 = 1/6 8

9. For example – 10 coloured beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, colour noted, returned to bag, then a second taken. 2nd 1st RR R B RB R G RG INDEPENDENT EVENTS R BR BB B B G BG R GR G GB B G GG

10. All ADD UP to 1.0 Probabilities 2nd 1st RR R P(RR) = 0.3x0.3 = 0.09 0.3 0.2 B RB P(RB) = 0.3x0.2 = 0.06 R G 0.3 0.5 RG P(RG) = 0.3x0.5 = 0.15 R BR P(BR) = 0.2x0.3 = 0.06 0.3 0.2 0.2 BB P(BB) = 0.2x0.2 = 0.04 B B 0.5 G BG P(BG) = 0.2x0.5 = 0.10 R GR P(GR) = 0.5x0.3 = 0.15 0.3 0.5 G 0.2 GB P(GB) = 0.5x0.2 = 0.10 B G GG P(GG) = 0.5x0.5 = 0.25 0.5

11. An aside: probability of multiple events Assume you have a 5/6 chance for an event to happen Rolling a 1-5 on a die, for example What’s the chance of that event happening twice in a row? Cases: Event happening neither time: 1/6 * 1/6 = 1/36 Event happening first time: 5/6 * 1/6 = 5/36 Event happening second time: 1/6 * 5/6 = 5/36 Event happening both times: 5/6 * 5/6 = 25/36 For an event to happen twice, the probabilityis the product of the individual probabilities 11

12. Probability vs. odds Consider an event that has a 1 in 3 chance of happening Probability is 0.333 Which is a 1 in 3 chance Or 2:1 odds Meaning if you play it 3 (2+1) times, you will lose 2 times for every 1 time you win This, if you have x:y odds, you probability is y/(x+y) The y is usually 1, and the x is scaled appropriately For example 2.2:1 That probability is 1/(1+2.2) = 1/3.2 = 0.313 1:1 odds means that you will lose as many times as you win 12

13. If you excel in probability then…..

15. CREDIT HAMIM HILAL – INFO REYNOLD JOJOE – INFO PARANTAP TRIPATHI – PPT KANESHKA GUPTA – PPT VARGHESE THOMAS – PIC/VID X - D