Probability and statistics, Year 11: dependent events, conditional probability and box plots

5/14

提示After the first red counter has gone, count what is left in the bag before writing the second fraction.

为什么First red: 5/8. Now 4 red remain out of 7, so second red: 4/7. Both: 5/8 × 4/7 = 20/56 = 5/14.

8/15

提示There are two orders in which one of each colour can come out.

为什么Red then blue: 4/6 × 2/5 = 8/30. Blue then red: 2/6 × 4/5 = 8/30. Add the two paths: 16/30 = 8/15.

Dependent events

提示Taking two sweets from a bag without putting the first back is the classic example; the second pick relies on the first.

为什么On a tree diagram, dependent events show up as second-stage branches whose probabilities differ according to which first branch was followed. To find 'A and B', multiply P(A) by the probability of B once A has happened.

13/15

提示Work out the one case that fails, no milk at all, and subtract it from 1.

为什么P(both dark) = 4/10 × 3/9 = 12/90 = 2/15. So P(at least one milk) = 1 − 2/15 = 13/15.

0.15; 0.07; 0.22

提示Multiply along each path that ends in 'late', then add the two results.

为什么Rain and late: 0.3 × 0.5 = 0.15. No rain and late: 0.7 × 0.1 = 0.07. These are the two separate ways of being late, so P(late) = 0.15 + 0.07 = 0.22.

7/16

提示Only one group in the table is in play — find its total first.

为什么The pupil is known to bring a packed lunch, so the choice is among those 9 + 7 = 16 pupils only. 7 of them are girls.

1/3

提示On a Venn diagram, look only inside the French circle.

为什么Because the student is known to study French, the choice is among the 12 in the French circle. 4 of them are in the overlap, so the probability is 4/12 = 1/3.

15/22

提示Count every day she was late, rainy or dry; that is the group being chosen from.

为什么She was late on 15 + 7 = 22 days, and 15 of those were rainy, so P(rainy given late) = 15/22. Working with expected frequencies out of 100 turns a tree diagram into simple counting.

A conditional probability

提示Questions about it usually contain the words 'given that'.

为什么Knowing that one event has happened shrinks the set of possible outcomes, so the total you divide by changes. In a two-way table it means using one row or column total instead of the grand total.

independent

提示It is the word for two events that have no effect on each other's chances.

为什么For dependent events the conditional probability differs from the plain one: the chance of a second red counter changes once a first has been removed. Comparing P(A given B) with P(A) is a test for independence.

A box plot

提示It has a rectangle in the middle with a line sticking out at each end; it is sometimes called a box-and-whisker diagram.

为什么The box runs from the lower quartile to the upper quartile, with a line inside it at the median. The whiskers reach out to the lowest and highest values.

15

提示The two quartiles are the two ends of the box.

为什么Lower quartile 20, upper quartile 35, so the interquartile range is 35 − 20 = 15. The range would be 50 − 12 = 38, and the median is 28.

Class B scored higher on average (higher median); class A's marks were more consistent (smaller interquartile range)

提示Make one statement about a typical score and one about how spread out the scores are.

为什么A full comparison uses one measure of average and one measure of spread, and says what each means for the data. The median gives the typical mark; the interquartile range shows how varied the middle half of the marks is.

40

提示The quartiles cut the data into four equal-sized groups.

为什么A quarter of the values lie below the lower quartile and a quarter above the upper quartile, so the box holds the middle half: 80 ÷ 2 = 40 plants. Each whisker covers about 20.

The median; the range and the interquartile range

提示Think about what the five marked values are, and what you get by subtracting pairs of them.

为什么The line in the box is the median. The range is the distance between the whisker ends, and the interquartile range is the length of the box. The mean and the mode cannot be found, because the individual values are not shown.