Functional Skills Mathematics

Transcription

1 Functional Skills Mathematics Level Learning Resource Probability D/L.

2 Contents Independent Events D/L. Page - Combined Events D/L. Page - 9 West Nottinghamshire College

3 D/L. Information Independent Events Probability is a measure of how likely an event is to happen. his is measured on a scale from 0 (impossible) to (certain) and can be represented by a fraction showing possible events compared to total number of all possible results. Possible events. otal number of all possible results he nearer the fraction is to, the more likely the event will happen. If you bought 0 raffle tickets and only 00 were sold you would have 0 chances out of 00 to win a prize. If you bought 0 raffle tickets and only 00 were sold you would have 0 chances out of 00 to win a prize. Obviously the more tickets you buy, the higher the probability that you will win. Probability can t tell us the outcome, it simply tell us how likely an event is to happen. Example What is the probability of a coin landing heads up when tossed? If you toss a coin it may land with the head side uppermost, or the tail side uppermost. here is a choice of outcomes so the chance of getting one of those outcomes is always in. No matter how many times you toss the coin, you will still have just a in chance off it landing heads up. Possible events. chance in otal number of all possible results West Nottinghamshire College

4 D/L. Examples Independent Events You have a set of different playing cards. If you are asked to select one card what are the chances that you will select the ace of spades? here is only ace of spades in the pack of cards so you have chance of picking the ace of spades and chances of picking another card i.e. You have a in chance of selecting a particular card. owever many times you repeat this, if you begin with a full set of different cards, you will have a in chance of choosing a particular card. Possible events. chance in otal number of all possible results You have a set of different playing cards. If you are asked to select one card, what is the probability that you will choose an ace card? here are aces in the pack so you have chances of picking an ace and 8 chances of picking another card. i.e. You have in ( in ) chance of picking an ace card. Possible events. chance in otal number of all possible results If you throw a six-sided die, what is the probability that you will throw a? here is only one on the die. When you throw the die there are possible results. So you have a in chance that you will throw a. Possible events. chance in otal number of all possible results If you throw a six-sided die, what is the probability that you will throw an odd number? here are odd numbers on a die (,, ). When you throw the die there are possible results. So you have a in ( in ) chance that you will throw a. Possible events. chance in otal number of all possible results West Nottinghamshire College

5 D/L. Exercise Independent Events Show all answers to the following questions as simple fractions. ) If you draw a card from a full deck of playing cards what is the chance that it will be: a) the ace of spades? b) the queen of hearts? c) a king? (here are kings in a pack of cards.) d) a picture card? (here are picture cards in a full pack.) ) Abigail has 7 chocolates left. As she offers her friend a chocolate she realises there are now only of her favourite truffles left. What is the probability that her friend will choose one of her favourites if she chooses a chocolate at random? ) An adult education class has eight women and six men on the register. What is the probability that the first person on the register is a woman? ) here are 00 employees in the building and will be chosen at random to go on a course. ow likely is it that Fern s name will be drawn out? ) Jonah has sisters and brothers. What is the likelihood that the person trying to get into his bedroom now is one of his brothers? ) here are 9 balls in the lottery draw. What chance is there that the first ball drawn out is one of your numbers? 0 8 West Nottinghamshire College

6 D/L. Information Combined Events Where two or more events are involved, a tree diagram can be used to help solve probability problems. he possible outcomes for each event are represented by branches, then each branch produces further outcomes depending on the individual events. Examples Example Ryan has a game at the school fête. A contestant has to toss a coin times. A prize is won if the contestant throws either heads in a row or tails in a row. st go nd go rd go Outcome We have already seen that each outcome is equally likely, because every time a coin is tossed there is a in probability of getting a head. here are 8 outcomes altogether. ow many ways are there of winning? here are ways of winning. So the probability of a win is 8 or. West Nottinghamshire College

7 D/L. Examples continued Combined Events Example If you throw one die, you have a in chance of getting a particular number. If you throw another die, you still have a in chance of getting a particular number. If you throw both dice together, what are the chances you ll throw sixes (i.e. )? Second die First die You have a in chance i.e. there are possible outcomes, only one of which is to throw sixes (totalling ). West Nottinghamshire College 7

8 D/L. Example Combined Events If you pick balls at random from a bag containing 7 red balls and 9 blue balls, what is the probability of picking red ball and blue ball? First pick Second pick Red RR 7 x = Red Blue 9 7 Blue Red RB = BR = 7 9 x = x = 0 8 Blue BB = x = 0 here are balls in total in the bag. On the first pick you have a 7 in chance of picking out a red ball. If you pick out a red ball, there are then just balls left in the bag. In the second pick there are only red balls left so you have a in chance of picking another red ball. here are still 9 blue balls in the bag so you have a 9 in chance of picking a blue ball. You have a 9 in chance of picking out a blue ball on the first pick. If you pick a blue ball, there will then be 8 blue balls left so there is an 8 in chance of picking another blue ball. here will still be 7 red balls in the bag for the second pick, so you have a 7 in chance of picking a red ball. Calculate the overall probability of an outcome by multiplying together the probabilities along the branches as shown above. here are outcomes which satisfy the condition required i.e. RB and BR give the result red ball and blue ball. Where there are several outcomes that satisfy the condition, the probabilities are added together to get the answer. + = = NB. If you add all the results together they add up to. his is a good check that your 7 0 calculation is correct = = West Nottinghamshire College 8

9 D/L. Exercise Combined Events ) What is the probability of Ryan getting heads in Example? ) Using the tree diagram shown in Example, complete the table to show the possible outcomes of throwing dice together, i.e. + = etc. ) From your table, you will see that there are possible outcomes. he probability of getting is (or 8 ) because there are two outcomes giving the answer. What is the probability of getting: a) b) c) 8 d) ) What outcome is most likely? ) Using Example as a guide, draw tree diagrams to show all the possible outcomes when two balls are picked from a bag containing: a) yellow and green balls; b) blue and pink balls; c) red and white balls. West Nottinghamshire College 9