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Geometry

Geometry. Probability of Compound Events. A single event is called a simple event . These events have fairly simple probabilities. Chance of rain next Saturday in Thousand Oaks 20%. Chance of rain next Saturday in Chicago 60%. 0.6. P(Rain in Chicago) =. 0.2. P(Rain in T.O) =.

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Geometry

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  1. Geometry Probability of Compound Events

  2. A single event is called a simple event. These events have fairly simple probabilities. Chance of rain next Saturday in Thousand Oaks 20% Chance of rain next Saturday in Chicago 60% 0.6 P(Rain in Chicago) = 0.2 P(Rain in T.O) = When flipping a coin twice (or flipping two coins at the same time) having a result of Tails on the first coin toss and Heads on the second is a simple event. However getting one Tail and one Head is a compound event as there are two ways in which that can happen. Look the Sample Space: {HH, TH, HT, TT}

  3. The weather in Thousand Oaks, CA, doesn’t affect the weather in Chicago, IL. These two events are called independent eventsbecause the outcome of one doesn’t affect the outcome of the other one. Similarly the result of one coin toss does not effect the result of another toss.

  4. What if we looked at the probability of rain occurring in both cities on Saturday? When two or more simple events are combined, it is considered a compound event. (Like the HT and TH outcomes when tossing two coins form the compound event of tossing one Head and one Tail) Probability of Compound Events If 2 events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B.

  5. The probability of rain occurring Saturday in both T.O. and Chicago is 12%.

  6. The event of it not raining in Chicago next Saturday is a special kind of event called a Complement. In this case it is the complement of it raining in Chicago next Saturday. You can think of Complements as two events that are opposites in terms of all the possibilities in the sample space. Example: When rolling a single die, the complement of rolling a 3 is the event of rolling a 1, 2, 4, 5 or 6.

  7. If the chance of rain in Chicago is 60%, then the chance of it not raining there is 40%. The probability of rain occurring Saturday in T.O. and not in Chicago is 8%.

  8. Andrew is flying from Birmingham to Chicago. On the first leg of the trip he has to fly from Birmingham to Houston. In Houston he’ll change planes and head to Chicago. Airline statistics report that the Birmingham to Houston flight has a 90% on-time record and the flight from Houston to Chicago has a 50% on-time record. Assuming that one flight’s on-time status is independent of another, what’s the probability that both flights will be on time? The probability of both flights being on time is 45%.

  9. A die is rolled and a spinner like the one shown on the right is spun. Find each probability: A B D C These are independent events. Multiply the probabilities together to find the probability of both occurring.

  10. A die is rolled and a spinner like the one shown on the right is spun. Find each probability: A B D C

  11. Find the probability that you’ll roll a 6 and then a five when you roll a die twice. These are independent events. You will multiply the probabilities together.

  12. Round plastic chips numbered 1-15 are placed in a box. Chips numbered 11-25 are placed in another box. A chip is randomly drawn from the box on the left, then a second chip is randomly drawn from the box on the right. These are independent events. Multiply the probabilities together. 1-15 11-25

  13. When the outcome of one event affects the outcome of another event, the two events are said to be dependent. Probability of Dependent Events If 2 events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B given A occurs. This is called a Conditional Probability. This can also be written using the formal notation for conditional probability: P(A and B)=P(A)•P(B|A)

  14. Here’s an example of two events that are dependent. A bag contains 2 green, 9 brown, 7 yellow, and 4 blue marbles. Once a marble is selected, it is not replaced. Find the probability of randomly drawing a brown, then a yellow marble. P(yellow following brown) = So, putting all that all together, we see that: P(brown and yellow) = P(brown)•P(yellow following brown) =P(brown)•P(yellow|brown)

  15. Start again with a bag that contains 2 green, 9 brown, 7 yellow, and 4 blue marbles. Again, once a marble is selected, it is not replaced. Find the probability of randomly drawing a green marble, then a marble that’s not blue. P(not blue|green) = P(green and not blue)

  16. Find the probability of randomly drawing a yellow marble, a yellow marble, and a blue marble. As before, marbles that are drawn are not replaced. P(yellow|yellow)= P(blue|yellow and yellow)= P(yellow and yellow and blue)

  17. At a school carnival, winners in a ring-toss game are randomly given a prize from a bag that contains 4 sunglasses, 6 hairbrushes, and 5 key chains. Three prizes are randomly chosen from the bag and not replaced. Find the probability: P(sunglasses and hairbrush and key chain) =P(sunglasses)•P(hairbrush|sunglasses)•P(key chain|sunglasses and hairbrush) 3 3 7

  18. Day 2 Probability of Mutually Exclusive and Inclusive Events

  19. Mutually Exclusive Events Mutually exclusive events, or disjoint events, are events that cannotoccur at the same time. For example, consider the events of rolling a 1 or a 3 on a die? A die can’t show both a 1 and a 3 at the same time, so the two events are considered mutually exclusive. Probability of Mutually Exclusive Events If 2 events, A and B, are mutually exclusive, then the probability that either A orB occurs is the sum of their probabilities.

  20. These events are mutually exclusive. The die can’t show two different numbers at the same time. This problem asks what the probability is of one or the other number showing. In other words, an outcome of 1 or an outcome of 3 is a desired outcome. The two probabilities must be added together to show that either roll is acceptable. Find the probability of rolling a 1 or a 3 on a die.

  21. Janet is going to an animal shelter to choose a new pet. The shelter has 8 dogs, 7 cats, and 5 rabbits available for adoption. If she randomly picks an animal to adopt, what is the probability it will be a cat or a dog? The problem asks for a cat or a dog, so either outcome is desired. The two probabilities should be added together.

  22. Bart randomly drew one card from a standard deck of cards. What is the probability that the card he drew was a club or a diamond? Bart randomly drew one card from a standard deck of cards. What is the probability that the card he drew was a face card or a 10?

  23. Mutually Inclusive Events Mutually inclusive events are events that canoccur at the same time. For example, what is the probability of drawing either an ace or a spade randomly from a deck of cards? It’s possible for a card to be both an ace and a spade at the same time. When we consider the probability of drawing either an ace or a spade there are 4 cards in the deck that are aces, 13 cards in the deck that are spades, and 1 card that’s both an ace and a spade at the same time. That one card should not be counted twice!

  24. Probability of Mutually Inclusive Events If 2 events, A and B, are mutually inclusive, then the probability that either A orB occurs is the sum of their probabilities, decreased by the probability of both occurring.

  25. Let’s go back to the earlier question: What is the probability of drawing either an ace or a spade? To figure this out you need to add the probability of drawing an ace to the probability of drawing a spade, then subtract out the probability of drawing a card that’s both an ace and a spade at the same time. P(ace or spade)  P(ace)  P(spade) P(ace and spade) =

  26. What is the probability of drawing either a red card or an ace? To figure this out you need to add the probability of drawing a red card to the probability of drawing an ace, then subtract out the probability of drawing a card that’s both red and an ace at the same time.

  27. What is the probability of drawing a face card or a spade?

  28. Suppose your dog had 9 puppies! • 3 are brown females • 2 are brown males • 1 is a mixed color female • 3 are mixed color males If a puppy is randomly chosen from the litter, what is the probability that it will be male or be mixed color? The puppies that are both male and mixed will be counted twice if not subtracted here.

  29. In a bingo game, balls or tiles are numbered from 1 to 75. The numbers correspond to columns on a bingo card. B I N G O 1-15 16-30 31-45 46-60 61-75 If a number is selected at random, what is the probability it is a multiple of 5 or in the “N” column? What multiples of 5 appear in the “N” column?

  30. The numbers that are both multiples of 5 and appear in the “N” column will be counted twice if they’re not subtracted out here.

  31. The numbers that are both even and appear in the “G” column will be counted twice if they’re not subtracted out here.

  32. Students are selected at random from a group of 12 boys and 12 girls. In that group there are 4 boys and 4 girls from each of 6th, 7th, and 8th grades. Find the probability:

  33. Students are selected at random from a group of 12 boys and 12 girls. In that group there are 4 boys and 4 girls from each of 6th, 7th, and 8th grades. Find the probability:

  34. To sum it all up for today: Mutually exclusive events are events that cannot occur at the same time. You will see the word “or” in the question. Add the probabilities of mutually exclusive events together to consider the probability that either one or the other will occur. Mutually inclusive events are events that can occur at the same time. You will see the word “or” in the question here too. Add the probabilities of mutually inclusive events together to consider the probability that either one or the other will occur, but remember to subtract out the events that overlap so they’re not counted twice.

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