S2: PROBABILITY (NOTES)

STATISTICS AND PROBABILITY 4052 ELEMENTARY MATHEMATICS S2: PROBABILITY CONTENT ● probability as a measure of chance ● probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability) ● probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate) ● addition and multiplication of probabilities (mutually exclusive events and independent events) PROBABILITY AS A MEASURE OF CHANCE ● Probability is a measure of how likely an event is to occur. The probability of an event is a number between 0 and 1, where: ○ 0 means the event will not occur. ○ 1 means the event will definitely occur. Formula for the probability of single event A: 𝑃 ( 𝐴 ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 . EXAMPLE: SIX-SIDED DIE If you roll a six-sided die, the probability of rolling a 3 is . One favourable 𝑃 ( 3 ) = 1 6 outcome and six possible outcomes. PROBABILITY OF SINGLE EVENTS LISTING ALL THE POSSIBLE OUTCOMES (SIMPLE CHANCE SITUATIONS) EXAMPLE: TOSSING A COIN Possible outcomes = {Heads, Tails} Total outcomes = 2 Probability of Heads = 1 2 Probability of Tails = 1 2 EXAMPLE: ROLLING A DIE AND GETTING AN EVEN NUMBER Possible outcomes = {1, 2, 3, 4, 5, 6} Total outcomes = 6 Probability of rolling an even number = = 3 6 1 2 EXAMPLE: SELECTING A DAY OF THE WEEK THAT FALLS ON A WEEKEND Possible outcomes = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} Total outcomes = 7 Probability of selecting a weekend day = 2 7 1 @BOOKSBFRBOYS | TIKTOK

STATISTICS AND PROBABILITY 4052 ELEMENTARY MATHEMATICS PROBABILITY OF SIMPLE COMBINED EVENTS POSSIBILITY DIAGRAMS ● Possibility Diagrams are useful when you want to show all the possible outcomes of a situation, particularly when events are sequential. EXAMPLE: TOSSING A COIN AND ROLLING A DIE Two fair six-sided dice are rolled. Find the probability that the sum of the numbers showing on the two dice is an odd number greater than 5, giving your answer as a fraction in simplest form. P(odd numbers greater than 5) = 12 36 = 1 3 TREE DIAGRAMS ● Tree Diagrams are helpful for understanding the probabilities of different sequences of events. Each branch of the tree represents a possible outcome, and the probabilities are multiplied along the branches. EXAMPLE: TOSSING A COIN AND ROLLING A DIE 1st event: rolling a die, {1, 2, 3, 4, 5, 6} 2nd event: tossing a coin, {Heads, Tails} P(2 and Heads) = 1 6 × 1 2 = 1 12 2 @BOOKSBFRBOYS | TIKTOK

STATISTICS AND PROBABILITY 4052 ELEMENTARY MATHEMATICS ADDITION AND MULTIPLICATION OF PROBABILITIES ADDITION RULE (FOR MUTUALLY EXCLUSIVE EVENTS) ● Mutually exclusive events are events that cannot happen at the same time . If and are mutually exclusive, then: 𝐴 𝐵 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) EXAMPLE A fair die is rolled. The probability of getting a is , and the probability of getting a 2 𝑃 ( 𝐴 ) [ ] 1 6 is . Since getting a 2 and getting a 5 5 𝑃 ( 𝐵 ) [ ] 1 6 is mutually exclusive: 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 1 6 + 1 6 = 2 6 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 1 3 MULTIPLICATION RULE ( INDEPENDENT EVENTS) ● Independent events are events where the occurrence of one event does not affect the probability of the other . If 𝐴and are independent, then: 𝐵 𝑃 ( 𝐴 𝑎𝑛𝑑 𝐵 ) = 𝑃 ( 𝐴 ) × 𝑃 ( 𝐵 ) EXAMPLE You flip a fair coin and roll a fair die. The probability of getting heads is , and 𝑃 ( 𝐴 )[ ] 1 2 the probability of rolling a is . 6 𝑃 ( 𝐵 ) [ ] 1 6 𝑃 ( 𝐴 𝑎𝑛𝑑 𝐵 ) = 1 2 × 1 6 = 1 12 GENERAL ADDITION RULE (FOR NON-MUTUALLY EXCLUSIVE EVENTS) ● If two events can happen at the same time, we use: 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) − 𝑃 ( 𝐴 𝑎𝑛𝑑 𝐵 ) EXAMPLE A card is drawn from a deck. Let be the 𝐴event of drawing a heart and B be 𝑃 ( 𝐴 ) = 13 52 ⎡ ⎣ ⎤ ⎦the event of drawing a face card . 𝑃 ( 𝐵 ) = 12 52 ⎡ ⎣ ⎤ ⎦Since 3 of these face cards are also hearts, 𝑃 ( 𝐴 𝑎𝑛𝑑 𝐵 ) = 3 52 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 13 52 + 12 52 − 3 52 = 22 52 𝑃 ( 𝐴 𝑜𝑟 𝐵 ) = 11 26 3 @BOOKSBFRBOYS | TIKTOK