This Checkpoint Task should be used in conjunction with the KS4–Core MathsConditional Probability Transition Guide.

Student Activity Sheets

Task 1

In groups, discuss the following statements and identify any misconceptions or mistakes made.
AI have spun a fair coin 6 times and got heads each time. It is more likely to be tails next time I spin the coin. / BTeam X plays Team B. Team X can win or lose or draw so the probability that they win is .
CIt is harder to throw a 6 than a 3 with a dice. / DThere are 2 blue counters and 3 red counters in a bag. If I pick a counter at random the probability that it is blue is .
ETomorrow it will either rain or not rain, so the probability that it rains is 0.5. / FIf I throw six fair dice at once, I am more likely to get 1 2 3 4 5 6 than 6 6 6 6 6 6
GI roll two fair six-sided dice and add the scores.The probability of getting 7 is because there are 11 possible scores I could get and 7 is one of them. / HMrs Smith has an operation. 80% of people who have the operation recover completely, therefore the probability that Mrs Smith recovers completely is 80%.
ITom buys some raffle tickets. He is more likely to win if he chooses numbers at random than if he picks numbers next to one another. / JI knew a man who smoked 40 cigarettes a day and lived to be 100, so smoking can’t be bad for you.
KA family has two children. The probability that they are both boys is
because they could be two girls, two boys or one of each. / L7 is a lucky number so you are more likely to win a raffle if you have the number 7.
M
Set A Set B

Set B has more black beads than set A so you are more likely to pick a black bead from set B than from set A. / N
The probability of getting white is for each spinner.

Task 2

Finding probabilities
  1. Draw a sample space diagram to show the outcome of rolling two fair dice. Use it to find the probability of

a)Two sixes

b)Rolling a 1 and a 5

c)A total of 7 when the scores are added

  1. If you flip three fair coins, what is the probability that you get two heads and a tail?
  1. There is a saying that, given an enormously long time and a typewriter, a tribe of monkeys would type the complete works of Shakespeare just by hitting the keys at random. What if the typewriter had just four keys: W H E N? List all possible arrangements of the letters and use this to find the probability of typing WHEN by hitting the keys at random if each letter can only be typed once.
Calculating with probabilities
  1. On the way to work Beverley passes through three sets of traffic lights. The probability that the first set is green when she reaches them is 0.6. The probability that the second set is green is 0.7 and the probability for the third is 0.8. What is the probability that she has to:

a)stop at all three sets of lights

b)stop at only one set of lights

c)stop at least two sets of lights

  1. There are nine boys and fifteen girls in a class. Three children are selected at random to represent the class in a competition. What is the probability that

a)all three are girls

b)one is a girl and two are boys

c)at least two girls are chosen

Conditional Probability
  1. Caroline’s Couture shop and Maggie’s market stall both sell a particular style of sweatshirt. Caroline accounts for 70% of these sweatshirt sales and Maggie accounts for the remaining 30%. Unfortunately colour dye that fades prematurely was used to manufacture the first batch of the product and the supplier estimates that 15% of the stock supplied to Caroline and 25% of the stock supplied to Maggie have this problem.

a)I can’t remember where I bought my sweatshirt. What is the probability that the colour of my sweatshirt will fade prematurely?

b)Given that my sweatshirt does fade prematurely, what is the probability that I purchased it from Maggie?

  1. The probability that a parachutists lands in a target area depends on the weather conditions. If it is windy, then the probability of landing in the target area is 2/5 but if it is not windy the probability is 4/5. The probability that it is windy on any random day in June is 1/6.

a)Calculate the probability that when the parachutist jumps on a random day in June he hits the target area.

b)Given that the parachutist landed in the target area on 15th June last year, calculate the probability that June 15th was a windy day.

Extension Task

  1. In 1986, University of Florida sociologist Michael Radelet claimed that if you killed a white person in Florida, the chances of getting the death penalty was three times greater than if you killed a black person. He classified 326 murders by race of victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person. If a victim from this study was white, what is the probability that that their murderer received the death sentence?. Do you agree with Radelet?
  1. An insurance company offers quotations for motor insurance by phone. 35% of quotations are provided by temporary agency staff, the rest by permanent staff. Unfortunately 22% of quotations provided by temporary staff are found to be wrong compared with 8% of quotations by permanent staff. Under the terms of the contract, the agency will pay a proportion of the costs associated with incorrect quotations based on the proportion of mistakes made by temporary staff. Find the probability that if a mistake has been made it has been made by one of the temporary staff and use your answer to suggest what proportion of the costs the agency should pay.
  1. Fred is a beagle “sniffer” dog at a cargo handling depot. Fred is 95% reliable in detecting contraband substances when they are present, and also has a probability of only 0.005 of indicating the presence of contraband substances when they are not present.

(i)If Fred indicates that contraband substances are present in 1% of cargoes he inspects, show that the probability, p, that a cargo contains contraband substances is 0.0053.

(ii)Using this value of p, obtain the probability of contraband substances in a cargo if Fred indicates their presence.

(iii)Fred’s younger brother Pete is still being trained. Pete is currently 90% reliable in detecting drugs when they are present, and has a probability of 0.01 of indicating the presence of drugs when they are not present. Fred and Pete share the work.
A cargo is investigated further if a dog indicates the presence of drugs.
What is the largest proportion of the work Pete should be allowed to do if no more than 1.2% of the cargoes are to be investigated further?
Again use the value p = 0.0053

  1. Here are two variations of the Monty Hall problem.

(a)Suppose that everything is the same except that Monty forgot to find out in advance which door has the car behind it. In the spirit of “the show must go on", he makes a guess at which of the two doors to open and gets lucky, opening a door behind which stands a goat. Now should the contestant switch or stick?

(b)You have observed the show for a long time and found that the car is put behind door A 45% of the time, behind door B 40% of the time and behind door C 15% of the time. Assume that everything else about the show is the same and that you pick door A. Monty opens a door with a goat and offers to let you switch. Should you? Suppose you knew in advance that Monty was going to give you a chance to switch. Should you have initially chosen door A?

March 2016