When you roll a die what are Your possible outcomes?

When you roll a die what are Your possible outcomes? Sample Space: S = {1, 2, 3, 4, 5, 6}

What are the probabilities For the possible outcomes? P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6

Lets say I rolled a 4 on my First roll. If I roll again what are my possible outcomes? Sample Space: S = {1, 2, 3, 4, 5, 6}

What are the probabilities For the possible outcomes? P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6

Sue is in line for an in-store give-away. There are four $10 gift cards, six $5 gift cards, six 10% off coupons, and one40% off coupon. What are Sue’s possible outcomes? Sample space: S = {$10 gift card, $10 gift card, $10 gift card, $10 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, 10% off coupon, 10% off coupon, 10% off coupon, 10% off coupon, 10% off coupon, 10% off coupon, 40% off coupon} = 17 outcomes

What are the probabilities of Sue’s possible outcomes? P($10 gift card) = 4/17 P($5 gift card) = 6/17 P(10% off coupon) = 6/17 P(40% off coupon) = 1/17

Let’s say that Sue got a 10% off coupon and Billy is in line next. Now, what are Billy’s possible outcomes? Sample space: S = {$10 gift card, $10 gift card, $10 gift card, $10 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, $5 gift card, 10% off coupon, 10% off coupon, 10% off coupon, 10% off coupon, 10% off coupon, 40% off coupon} = 16 outcomes

What are the probabilities of Sue’s possible outcomes? P($10 gift card) = 4/16 P($5 gift card) = 6/16 P(10% off coupon) = 5/16 P(40% off coupon) = 1/16

Do the probabilities change At all after Sue gets her in-store Give-away? YES

Independent Events Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring

Dependent Events When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed

Dependent or independent? Drawing a card from a deck of cards, not replacing it, and then drawing a second card. Dependent Drawing a card from a deck of cards, replacing it, and then drawing a second card. Independent

Dependent or independent? Being a lifeguard and getting a tan Dependent Parking in a no-parking zone and getting a parking ticket. Dependent

Dependent or independent? You accidentally dropped a coin from the top of 10 stairs. What is the probability that it will land on the eighth step, tails up? Event A = { } Event B = { } Eighth step Tails Up

Dependent or independent? Event A = { } Event B = { } Eighth step Tails Up If the coin does land on the eighth step, does that change the probability that it will land tails up? No, therefore they are independent

Dependent or independent? There are 3 black cats and 5 grey cats in a cage, and none of them want to be in there. The cage door opens briefly and two cats escape. What is the probability that both escaped cats are black? Event A = { } Event B = { } Black Cat Black Cat

Dependent or independent? Event A = { } Event B = { } Black Cat Black cat If the first cat to escape is a black cat, does that change the probability that the second will be a black cat? Yes, therefore they are dependent

Conditional Probability The probability that event B occurs after A has already occurred. Symbol: P(B|A) (read “Probability of B given A”)

Conditional Probability Example Using a deck of cards, find the probability of drawing an Ace given that you have already drawn a Jack. Find: P(B|A) Let A = drawing a Jack Let B = drawing an Ace P(Ace|Jack)

Conditional Probability Example Find: P(B|A) “probability of drawing an Ace given that we have already drawn a Jack.” -We still have all possible Aces to draw from. -Now that we drew a Jack there are only 51 cards left in our deck of cards. _4_ 51 ____Aces_____ Cards remaining P(B|A) = =

Conditional Probability Example Given a hat with the numbers 1-10, find the probability of selecting a number greater than 4, given that you have already selected, and not replaced, the number 7. Find: P(B|A) = P(greater than 4|7) Let A = selecting the number 7 Let B = selecting a number greater than 4

Conditional Probability Example Find: P(B|A) “probability of selecting a number greater than 4 given that we have already selected the number 7.” -What numbers are still remaining? {1, 2, 3, 4, 5, 6, 8, 9, 10} -What of those numbers are greater than 4? { 5, 6, 8, 9, 10} _5_ 9 ___{5, 6, 8, 9, 10}_____ {1, 2, 3, 4, 5, 6, 8, 9, 10} P(B|A) = =

P(A and B) for Independent Events When two events are independent, the probability of both occurring is: P(A and B) = P(A) * P(B)

Example: P(A and B) for Independent Events Find the probability of rolling a die and getting a 6, then rolling again and getting a 4. Event A = { } Event B = { } Rolling a 6 Rolling a 4 Does event A have any affect on event B occurring? No, they are independent

Example: P(A and B) for Independent Events Find the probability of rolling a die and getting a 6, then rolling again and getting a 4. Event A = { } Event B = { } Rolling a 6 Rolling a 4 P(A and B) = P(A) * P(B) = P(6) * P(4) = 1/6 * 1/6 = 1/36

Example: P(A and B) for Independent Events Using a deck of cards, find the probability of getting an Ace, replacing it, and then getting a heart. Event A = { } Event B = { } Ace heart Does event A have any affect on event B occurring? No, they are independent

Example: P(A and B) for Independent Events Using a deck of cards, find the probability of getting an Ace, replacing it, and then getting a heart. Event A = { } Event B = { } Ace heart P(A and B) = P(A) * P(B) = P(Ace) * P(heart) = 4/52 * 13/52 = 1/52

P(A and B) for Dependent Events When two events are dependent, the probability of both occurring is: P(A and B) = P(A) * P(B|A)

Example: P(A and B) for Dependent Events Using a deck of cards and selecting two cards, find the probability that both are spades. Event A = { } Event B = { } Spade Spade Does event A have any affect on event B occurring? Yes, they are dependent

Example: P(A and B) for dependent Events Using a deck of cards and selecting two cards, find the probability that both are spades. After selecting a spade, there are only 12 remaining spades out of the 51 remaining cards Event A = { } Event B = { } Spade Spade P(A and B) = P(A) * P(B|A) = P(Spade) * P(Spade|Spade) = 13/52 * 12/51 = 1/17

Example: P(A and B) for Dependent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin? Event A = { } Event B = { } Annie Justin Does event A have any affect on event B occurring? Yes, they are dependent

Example: P(A and B) for dependent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin? Event A = { } Event B = { } Annie Justin P(A and B) = P(A) * P(B|A) = P(Annie) * P(Justin|Annie) = 1/12 * 1/11 = 1/132

If 18% of all Americans are underweight, find the probability that if three Americans are selected at random, all will be underweight. *When a small sample is selected from a large population and subjects are not replaced the probability changes so little that it is considered to stay the same.

If 18% of all Americans are underweight, find the probability that if three Americans are selected at random, all will be underweight. Event A = { } Event B = { } Event C = { } Underweight Underweight Underweight P(A and B and C) = P(A) * P(B) * P(C) = .18 * .18 * .18 = .005832

When you roll a die what are Your possible outcomes?