Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 1 • Question 27

Dom is taking part in a card tournament. He plays three games, and the probability that he wins each game is 0.3. Assume each game is independent. (a) Complete the tree diagram. To qualify for the next stage of the tournament, Dom needs to win at least two games. (b) Calculate the probability that after the third game, Dom qualifies for the next stage by winning exactly two games.

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Video Transcript

Dom is taking part in a card tournament. He plays three games. And the probability that he wins each game is 0.3. Assume each game is independent. Part a) Complete the tree diagram.

As each game is independent, the result of the first game has no impact on the results of the second game. Likewise, the result of the second game has no impact on the third game. The probability of winning any game will always be 0.3.

As there are only two possible outcomes, winning and losing, if the probability of Dom winning the game is 0.3, then the probability of him losing the game is one minus 0.3 as the probabilities must sum to one.

One minus 0.3 is equal to 0.7. Therefore, the probability of Dom losing any game is 0.7. This means that any branch in our tree diagram that says win will have a probability of 0.3 and any branch that says loss will have a probability of 0.7.

If Dom wins the first game, he can have a win or a loss on the second game. The probabilities on these branches will be 0.3 and 0.7, respectively. Likewise, if he lost the first game, he could also win or lose the second game. The probability of a win is 0.3 and the probability of a loss is 0.7.

We then repeat this process for the third game. Once again, the probability of winning any of these games is 0.3 and the probability of losing them is 0.7.

We now have our completed tree diagram, showing the eight possible outcomes over the three games: win, win, win, win, win, loss, win, loss, win, and so on all the way down to loss, loss, loss.

The second part of the question says the following.

To qualify for the next stage of the tournament, Dom needs to win at least two games. Part b) Calculate the probability that after the third game, Dom qualifies for the next stage by winning exactly two games.

We want to work out the probability that Dom wins exactly two out of his three games. This could happen in three ways. He could win the first game, win the second game, and lose the third game: win, win, lose. Or he could win the first game, lose the second game, and win the third game. Or he could lose the first game and win the second and third games.

In probability, the word “and” means multiply. In this case, Dom has to win the first game and the second game. And then, he has to lose the third game. Or he has to win the first game and lose the second game and win the third game. Or finally, he needs to lose the first game and win the second game and win the third game.

We need to multiply the probabilities of each event. As the probability of winning is 0.3 and the probability of losing is 0.7, we need to multiply 0.3 by 0.3 by 0.7.

To work out the probability of win, lose, win, we need to multiply 0.3 by 0.7 by 0.3. Finally, to work out the probability of lose, win, win, we need to multiply 0.7 by 0.3 by 0.3.

You will notice that all three of these calculations are the same as multiplication is commutative. It doesn’t matter which order we multiply the numbers.

As this is a non-calculator paper, we might wish to write these decimals as fractions. 0.3 is equal to three-tenths or three over 10 and 0.7 is equal to seven-tenths. We can then solve this calculation by multiplying the numerators three, three, and seven and multiplying the denominators 10, 10, and 10.

Three multiplied by three is equal to nine. And nine multiplied by seven is equal to 63. 10 multiplied by 10 is equal to 100. And 100 multiplied by 10 equals 1000.

This means that the probability of Dom winning the first game and winning the second game and losing the third game is 63 out of 1000. This is the same as 63 divided by 1000, which is equal to 0.063.

As mentioned previously, multiplication is commutative. Therefore, the probability of win, lose, win is also 0.063. Likewise, the probability of lose, win, win is 0.063.

There are three possible ways that Dom could win exactly two out of his three games. The word “or“ in probability means add. Therefore, we need to add these three answers.

As the three numbers are identical, we could also do this by multiplying 0.063 by three. 63 multiplied by three is equal to 189. Therefore, our probability is 0.189.

We can say that the probability of Dom qualifying by winning exactly two games is 0.189.

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