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Grade 8 Probability

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Grade 8 Probability

Show your working where needed. Give probabilities as simplified fractions or decimals. Assume every stated choice is equally likely unless the question says otherwise.

Core probability

Use favourable outcomes ÷ total outcomes when outcomes are equally likely.

1.A pouch holds 6 orange counters, 4 purple counters and 2 white counters. Find the probability of selecting a purple counter.

2.Using the same pouch, find the probability that a counter selected at random is not white.

3.A fair six-sided die is rolled. Find the probability of rolling (a) a number less than 3, (b) an odd number, and (c) a 2 or a 5.

4.One card is selected from a standard 52-card deck. Find the probability of drawing (a) a diamond, (b) an ace, (c) a black card, and (d) the ace of spades.

5.At a community sports club, 21 of 35 members play basketball. One member is chosen at random. Find the probability that the member plays basketball.

6.A fair spinner has equal sections numbered 1 to 10. Find the probability of landing on (a) a multiple of 4, and (b) a prime number.

7.Events C and D cannot happen on the same trial. If P(C) = 0.28 and P(D) = 0.37, find P(C or D).

8.A letter is selected at random from the word REGENERATION. Find the probability of selecting a vowel. Count each letter position separately.

9.The probability that a bus arrives late is 0.18. Find the probability that it does not arrive late.

10.A coin is tossed 240 times and lands on tails 132 times. Find the experimental probability of tails and compare it with the theoretical probability.

Sample spaces and multi-step events

Use an organised list or a tree diagram to keep track of outcomes.

11.A coin is tossed and then a fair four-sided die numbered 1 to 4 is rolled. How many outcomes are possible? List them and find the probability of getting heads and an even number.

12.A bag contains one yellow token and one black token. A token is drawn, replaced, then another token is drawn. List the ordered outcomes and find the probability that both draws show the same colour.

13.A container has 2 green counters and 2 red counters. Two counters are drawn without replacement. Find the probability of drawing one counter of each colour, in either order.

14.Three fair coins are tossed. How many ordered outcomes are there, and what is the probability of getting exactly one head?

15.Two fair six-sided dice are rolled. Find the probability that their sum is 8.

16.A café offers 4 sandwich fillings, including egg, and 3 drink choices, including apple juice. If a meal consists of one filling and one drink, how many different meals can be made? If each combination is equally likely, find the probability of choosing egg filling with apple juice.

17.A fair coin is tossed four times. Find the probability of getting exactly three heads.

18.A bag contains 4 blue and 3 orange counters. Two counters are drawn with replacement. Find the probability that the two counters are different colours.

Tables, sets and conditional probability

Use the information in each question; the table is included where needed.

19.A survey of 90 students found that 52 play netball, 38 play football and 17 play both. Complete the table and state how many play neither sport.

Netball

No netball

Total

Football

No football

Total

20.Use the completed sports table. Find (a) the probability a randomly chosen student plays netball but not football, (b) the probability a student plays both sports given that they play football, and (c) the probability a student plays neither sport.

21.A group of 32 students was surveyed: 18 enjoy art, 15 enjoy drama, and 7 enjoy both. Fill the four regions of a Venn diagram and find the number who enjoy neither.

22.For the art and drama survey, find (a) P(art or drama), (b) P(art and drama), and (c) P(not art).

23.In a survey of 60 people, 34 drink tea, 29 drink coffee, and 12 drink both. Find the probability that a person drinks exactly one of these beverages.

24.The table shows survey responses about walking for exercise and bringing a reusable water bottle. Find (a) P(brings a bottle), (b) P(walks | brings a bottle), and (c) P(brings a bottle and does not walk).

Walks

Does not walk

Total

Brings bottle

No bottle

Total

100

25.Using the walking and water-bottle table, determine whether walking and bringing a bottle are independent. Show a comparison that supports your conclusion.

Extension

These problems involve several steps. Explain your reasoning as well as giving the result.

26.A box contains 6 red, 4 blue and 2 yellow counters. Two counters are drawn without replacement. Find the probability that they are the same colour.

27.A prize is behind one of three doors. You choose Door A. The host, who knows where the prize is, opens Door C to reveal no prize. The prize was equally likely to be behind any door, and the host always opens an unchosen door without the prize. If both unchosen doors are safe, the host chooses randomly between them, with each door equally likely to be opened. Should you stay with Door A or switch to Door B? Explain.

28.Assume birthdays are equally likely across 365 days, with no leap year. For a group of 25 people, the probability that all birthdays are different is approximately 0.431. Use the complement to find the approximate probability that at least two people share a birthday.

29.A condition affects 2% of a population. A test correctly returns positive for 90% of people with the condition and also returns positive for 8% of people without it. If a person tests positive, find the probability they have the condition.

30.A standard deck has 52 cards. Let A be the event that a card is a club and B the event that it is a face card (jack, queen or king). Find (a) whether A and B are independent, (b) P(A or B), and (c) P(A given B).