1 / 23

Probability

Probability. 50-50 chance. likely. unlikely. poor chance. Independent Events. certain. possible. probable. Vocabulary. A sample space is the set of all possible outcomes . A simple example is flipping a coin. The sample space is {heads, tails}. Vocabulary.

ulani
Download Presentation

Probability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Probability 50-50 chance likely unlikely poor chance Independent Events certain possible probable

  2. Vocabulary A sample space is the set of all possible outcomes. A simple example is flipping a coin. The sample space is {heads, tails}

  3. Vocabulary Two events are independentif the outcome of one has no effect on the outcome of the other. Examples are rolling two dice, or spinning a spinner and rolling a dice

  4. Vocabulary Two events are dependentif the outcome of one event relies on the other event. Examples are picking two marbles out of a bag, or two socks out of a drawer

  5. 3 4 Vocabulary The complement of an event is all the outcomes NOT included in the event. It is shown by A’. The following spinner is spun once: What is the probability of it not landing on the yellow sector? P(not yellow) OR P(yellow’) =

  6. Vocabulary The intersection of two events is all the outcomes that are SHARED by both events. It is denoted by A∩B and can be read A and B Event A is even numbers on a dice Event B is multiples of 3 on a dice A ∩B is the number 6 P(A ∩B) = 1/6 If two events are independent, then their intersection can be calculated as P(A ∩B)=P(A)*P(B)

  7. Vocabulary The union of two events is all the outcomes of either event events. It is denoted by AUB and can be read A or B. Event A is odd numbers on a dice Event B is multiples of 3 on a dice A UB is {1,3,5,6} P(AUB) = 4/6 or 2/3 Union can be calculated as P(AUB)=P(A)+P(B)- P(A∩B)

  8. Consider a marriage or union of two people – when two people marry, what do they do with their possessions ? This symbol means “union” The bride takes all her stuff & the groom takes all his stuff & they put it together! And live happily ever after! This is similar to the union of A and B. All of A and all of B are put together!

  9. The two spinners are spun. What is the probability that both spinners will show an even number? P(1st spinner even) = 4 8 P(2nd spinner even) = 4 8 P(both spinners even) =4 ∙ 4 = 1 8 8 4 Example

  10. A game uses a dice and a spinner. A player rolls a dice. What is the P(odd #)? The player spins the spinner. What is the P(red)? 3. What is the P(odd # and Red)? Example

  11. There are 4 red, 8 yellow and 6 blue socks in a drawer. Once a sock is selected it is not replaced. Find the probability that 2 blue socks are chosen. P(1st blue sock) = 6 18 P(2nd blue sock) = 5 17 P(Two blue socks) = 6 ∙ 5 = 5 18 17 51 Example # of socks after 1 blue is removed Total # of socks after 1 blue is removed

  12. Example • In a certain town, the probability that a person plays sports is 65%. The probability that a person is between the ages of 12 and 18 is 40%. The probability that a person plays sports and is between the ages of 12 and 18 is 25%. Are the events independent?

  13. Outcomes are mutually exclusive if they cannot happen at the same time. Mutually exclusive outcomes For example, when you toss a single coin either it will land on heads or it will land on tails. There are two mutually exclusive outcomes. Outcome A: Head Outcome B: Tail

  14. Outcome A: the pupil has brown eyes. Outcome B: the pupil has blue eyes. Outcome C: the pupil has black hair. Outcome D: the pupil has wears glasses. A pupil is chosen at random from the class. Which of the following pairs of outcomes are mutually exclusive? Mutually exclusive outcomes These outcomes are mutually exclusive because a pupil can either have brown eyes, blue eyes or another colour of eyes. These outcomes are not mutually exclusive because a pupil could have both black hair and wear glasses.

  15. For example, a game is played with the following cards: 2 1 1 1 1 and P(sun) = 3 3 3 3 3 + = If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability. Adding mutually exclusive outcomes What is the probability that a card is a moon or a sun? P(moon) = Drawing a moon and drawing a sun are mutually exclusive outcomes so, P(moon or sun) = P(moon) + P(sun) =

  16. For example, a game is played with the following cards: 1 1 and P(star) = 3 3 If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability. Adding mutually exclusive outcomes What is the probability that a card is yellow or a star? P(yellow card) = Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star. P (Y U S) = P(Y) + P(S) – P(Y∩S) = 1/3 + 1/3 – 1/9 = 5/9.

  17. Venn Diagrams • Venn diagrams are useful in figuring out probabilities P(AUB)’

  18. Example • If A is the students who own bikes and B is the students who own skateboards, find • A∩B and P(A∩B). Are the events independent? • AUB and P(AUB) • (AUB)’ and P(AUB)’

  19. 20 The “eggs” show Math and Biology Ex. Amongst a group of 20 students, 7 are taking Math and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology? Solution: The diagram shows the 20 students. B M 4 3 3 do both 5 do neither 5 7 take Math ( but we have 3 already )

  20. 20 Amongst a group of 20 students, 7 are taking Maths and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology? Solution: The diagram shows the 20 students. B M The final number (taking Biology but not Math ) is given by 4 3 8 5 So, P( student takes Biology ) =

  21. Filling in a Venn diagram 100 people are asked if they eat meat, fish, both, or neither. You are told that 55 eat meat, 52 eat fish, and 21 eat neither. Use this information to complete the Venn diagram below. 21 eat neither 27 28 24 21

  22. Finding probabilities Use the Venn diagram to find the probability that someone picked at random: a) eats meat, b) eats fish, c) eats neither, d) eats only fish, e) eats both. ‘Given that’ Given that a man eats meat, find the probability that he also eats fish.

  23. The “and” rule Summary of methods The “or” rule • The word “or” often indicates that the probabilities need to be added together. • P(A or B) = P(AUB) = P(A) + P(B) – P(A∩B) • The word “and” often indicates that the probabilities need to be multiplied together. • P(A and B) = P(A∩B) = P(A) × P(B) if the two events are independent.

More Related