Every card in Probability and statistics, Year 11: dependent events, conditional probability and box plots
The whole deck, in order — so you can read it through before your child ever sees it.
- A bag holds 5 red and 3 blue counters. Two counters are taken out, one after the other, without replacement. What is the probability that both are red? Give a fraction in its simplest form.
5/14
HintAfter the first red counter has gone, count what is left in the bag before writing the second fraction.
WhyFirst red: 5/8. Now 4 red remain out of 7, so second red: 4/7. Both: 5/8 × 4/7 = 20/56 = 5/14.
- A bag holds 4 red and 2 blue counters. Two counters are taken out without replacement. What is the probability of getting one of each colour? Give a fraction in its simplest form.
8/15
HintThere are two orders in which one of each colour can come out.
WhyRed 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.
- Two events for which the outcome of the first changes the probability of the second
Dependent events
HintTaking two sweets from a bag without putting the first back is the classic example; the second pick relies on the first.
WhyOn 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.
- A box holds 6 milk chocolates and 4 dark chocolates. Two are taken at random, one after the other, and eaten. What is the probability that at least one of them is milk? Give a fraction in its simplest form.
13/15
HintWork out the one case that fails, no milk at all, and subtract it from 1.
WhyP(both dark) = 4/10 × 3/9 = 12/90 = 2/15. So P(at least one milk) = 1 − 2/15 = 13/15.
- The probability of rain is 0.3. If it rains, the probability that Mo is late is 0.5; if it does not rain, it is 0.1. So P(rain and late) = ____, P(no rain and late) = ____, and P(late) = ____.
0.15; 0.07; 0.22
HintMultiply along each path that ends in 'late', then add the two results.
WhyRain 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.
- A two-way table sorts the 40 pupils in a year group. 10 boys and 14 girls have a school lunch; 9 boys and 7 girls bring a packed lunch. A pupil is picked at random from those who bring a packed lunch. What is the probability that the pupil is a girl? Give a fraction in its simplest form.
7/16
HintOnly one group in the table is in play — find its total first.
WhyThe pupil is known to bring a packed lunch, so the choice is among those 9 + 7 = 16 pupils only. 7 of them are girls.
- In a class of 30 students, 12 study French and 10 study Spanish. 4 of these students study both languages and are included in both counts. A student who studies French is chosen at random. What is the probability that this student also studies Spanish? Give a fraction in its simplest form.
1/3
HintOn a Venn diagram, look only inside the French circle.
WhyBecause 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.
- Over 100 school days, it rains on 30. Zara is late on 15 of the 30 rainy days and on 7 of the 70 dry days. A day on which Zara was late is chosen at random. What is the probability that it was a rainy day? Give a fraction in its simplest form.
15/22
HintCount every day she was late, rainy or dry; that is the group being chosen from.
WhyShe 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.
- The probability of an event, worked out on the understanding that another event is already known to have happened
A conditional probability
HintQuestions about it usually contain the words 'given that'.
WhyKnowing 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.
- If P(A given B) is equal to P(A), then knowing that B has happened makes no difference to A, so A and B are ____.
independent
HintIt is the word for two events that have no effect on each other's chances.
WhyFor 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 diagram that shows a set of data by its lowest value, lower quartile, median, upper quartile and highest value, drawn against a scale
A box plot
HintIt has a rectangle in the middle with a line sticking out at each end; it is sometimes called a box-and-whisker diagram.
WhyThe 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.
- On a box plot, the box runs from 20 to 35, the line inside the box is at 28, and the whiskers end at 12 and 50. What is the interquartile range? (number only)
15
HintThe two quartiles are the two ends of the box.
WhyLower 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.
- Two box plots show the test marks of two classes. Class A has a median of 55 and an interquartile range of 10. Class B has a median of 62 and an interquartile range of 22. Write two comparisons of the classes in context.
Class B scored higher on average (higher median); class A's marks were more consistent (smaller interquartile range)
HintMake one statement about a typical score and one about how spread out the scores are.
WhyA 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.
- A box plot summarises the heights of 80 plants. About how many of the plants have heights that lie inside the box, between the lower quartile and the upper quartile? (number only)
40
HintThe quartiles cut the data into four equal-sized groups.
WhyA 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.
- Which measure of average can be read straight from a box plot, and which two measures of spread?
The median; the range and the interquartile range
HintThink about what the five marked values are, and what you get by subtracting pairs of them.
WhyThe 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.
1★ GCSE-MATH-PRB-0011
2★ GCSE-MATH-PRB-0012
3★ GCSE-MATH-PRB-0013
4★ GCSE-MATH-PRB-0014
5★ GCSE-MATH-PRB-0015
6★ GCSE-MATH-PRB-0016
7★ GCSE-MATH-PRB-0017
8★ GCSE-MATH-PRB-0018
9★ GCSE-MATH-PRB-0019
10★ GCSE-MATH-PRB-0020
11★ GCSE-MATH-PRB-0021
12★ GCSE-MATH-PRB-0022
13★ GCSE-MATH-PRB-0023
14★ GCSE-MATH-PRB-0024
15★ GCSE-MATH-PRB-0025
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