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Probability

Understand the concept of probability and learn how to estimate probabilities using relative frequency. Explore examples of mutually exclusive events, the probability of an event not happening, combining events, independent events, and tree diagrams.

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Probability

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  1. Probability The chance that something can happen

  2. Probability scale • A probability of 0 means that an event is impossible • A probability of 1 means that an event is certain • Probability of an event happening is given as a fraction or decimal • Probability of an event happening is written as P(x) where x is the event • P(x) = Number of possible outcomes in the event total number of possible outcomes

  3. Probability Examples • A fair dice is rolled. What is the Probability of getting: • a 6 • an odd number • a 2 or a 3? There are 6 possible outcomes. • P(6)=1/6 there is only one 6 • P(odd no.)=3/6 1,3,5 are odd numbers • P(2 or 3)=2/6 2 and 3 only appear once

  4. Examples • This table shows how 100 counters are coloured red or blue and numbered 1 or 2. The 100 counters are put in a bag and a counter is taken from the bag at random. • Calculate the probability that the counter is red? • Calculate the probability that the counter is blue and numbered 1?

  5. Counters continued Total number of counters = 23+32+19+26 = 100 Total number of red counters = 23+32 = 55 • P(Red)= 55/100 • P(Blue and 1)= 19/100

  6. Estimating Probabilities using Relative Frequency • The Relative Frequency of an event is given by: Relative Frequency= No. of times an event happens in an experiment(or survey) Total No. of trials in the experiment (or observations in survey)

  7. Relative Frequency Examples • In an experiment a pin is dropped for a 100 trials. The drawing pin lands “point up” 37 times. What is the relative frequency of the drawing pin landing “point up”? Relative Frequency = 37/100 = 0.37

  8. Relative Frequency Examples • A counter was taken from a bag of counters and replaced. The relative frequency of getting a blue counter was found to be 0.4. There are 20 counters in the bag. Estimate the number of blue counters. Relative frequency of getting a blue counter is 0.4. Total number of counters in bag = 20 Number of blue counters = 20 x 0.4 = 8. There are 8 blue counters in the bag.

  9. Mutually Exclusive Events • These are events that cannot happen at the same time. • For example when you throw a coin you cannot get a “Heads” at the same time as a “tails.” • When A and B are events which cannot happen at the same time: P(A or B) = P(A) + P(B)

  10. The probability of an event not happening • The events A and not A cannot happen at the same time. Because the events A and not A are certain to happen: P(not A) = 1 - A

  11. Example • A bag contains 3 red (R) counters, 2 blue (B) counters and 5 green (G) counters. A counter is taken from the bag at random. What is the probability that the counter is: • Red • Green • Red or green? Total number of counters = 3+2+5=10 • P(R) = 3/10 3 red counters • P(G) = 5/10=1/2 5 green counters • P(R or G) = 8/10 3 red +5 green

  12. Example • A bag contains 10 counters. 3 of the counters are red(R). A counter from the bag is taken at random. What is the probability that the counter is: • Red • Not red? • P(R) = 3/10 3 red counters out of 10 • P(not R) = 1- P(R) = 1- 3/10 = 7/10

  13. Combining 2 events • A fair coin is thrown twice. Write down all the possible outcomes and their probabilities. • A fair dice is thrown twice. Write down all the possible outcomes and find the probability of throwing a double.

  14. Independent Events • Independent events are events that do not influence eachother. When two coins are thrown one result does not effect the other or the person sitting next to you can swim but this does not effect your ability to swim or not swim. • When A and B are independent events then the probability of A and B occurring is given by P(A and B) = P(A) x P(B) P(A and B and C)= P(A) x P(B) x P(C)

  15. Tree diagrams • Box A contains 1 red ball (R) and 1 blue ball (B). Box B contains 3 red balls (R) and 2 blue balls (B). A ball is taken at random from Box A. A ball is then taken at random from Box B. • Draw a tree diagram to show all possible outcomes. • Calculate the probability that 2 red balls are taken.

  16. Tree Diagrams • The probability that Amanda is late for school is 0.4. Use a tree diagram to find the probability that on two days running • She is late twice • She is late exactly once

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