Probability, Year 8: more practice

1/2 (5 out of 10)

提示Count the favourable counters and all the counters.

为什么There are 3 + 5 + 2 = 10 counters in all, and 5 are blue: 5/10 = 1/2.

6

提示Fix the first dice and find the second that makes 7.

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

3/36 = 1/12

提示List the pairs that make 4 in a table of outcomes.

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

0.4

提示The three probabilities must add up to 1.

为什么1 − 0.35 − 0.25 = 0.4.

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

提示Add them together.

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

0.45

提示Experimental probability = successes ÷ number of trials.

为什么36 ÷ 80 = 0.45. This is an estimate: a different set of 80 tosses would give a slightly different answer.

Relative frequency

提示It is the experimental probability.

为什么Relative 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

提示Many trials make random variation small.

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

1/6

提示This is the theoretical probability.

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

6

提示Expected frequency = probability × number of trials.

为什么0.2 × 30 = 6. It is an expected value, not a promise.