Probability, Year 8: more practice

1/2 (5 out of 10)

HintCount the favourable counters and all the counters.

The whyThere are 3 + 5 + 2 = 10 counters in all, and 5 are blue: 5/10 = 1/2.

6

HintFix the first dice and find the second that makes 7.

The why(1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Seven is the most likely total.

3/36 = 1/12

HintList the pairs that make 4 in a table of outcomes.

The whyThe pairs are (1,3), (2,2) and (3,1), so 3 outcomes out of 36. 3/36 = 1/12.

0.4

HintThe three probabilities must add up to 1.

The why1 − 0.35 − 0.25 = 0.4.

No — they add up to 1.2, which is more than 1

HintAdd them together.

The whyThe probabilities of all the outcomes must total exactly 1. 0.5 + 0.3 + 0.4 = 1.2.

0.45

HintExperimental probability = successes ÷ number of trials.

The why36 ÷ 80 = 0.45. This is an estimate: a different set of 80 tosses would give a slightly different answer.

Relative frequency

HintIt is the experimental probability.

The whyRelative frequency is the fraction of trials in which the event happened. It is often used to estimate probability when theory is not available.

The dice may be biased

HintMany trials make random variation small.

The whyWith very large numbers of trials, the experimental probability should settle close to the theoretical value for a fair dice.

1/6

HintThis is the theoretical probability.

The whyThe more trials, the closer the experimental probability usually gets to the theoretical one. Short runs can be far off.

6

HintExpected frequency = probability × number of trials.

The why0.2 × 30 = 6. It is an expected value, not a promise.