Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
15 questions · official format, shuffled order
1 A fair coin is tossed and a fair die is rolled. What is the probability of a head and an even number?
Show worked solution
½ × ½ = 1/4. (1/12 counts only one even number, a single face out of 12 outcomes.)
2 A bag holds 4 red and 3 blue counters. Two are taken out, one after the other, without replacement. What is the probability that both are red?
Show worked solution
4/7 × 3/6 = 12/42 = 2/7. (16/49 puts the first counter back.)
3 The same bag (4 red, 3 blue) is used, but this time the first counter IS put back before the second is drawn. What is the probability that both are red?
Show worked solution
4/7 × 4/7 = 16/49. (2/7 is the without-replacement answer.)
4 A fair die is rolled twice. What is the probability of at least one six?
Show worked solution
1 − (5/6)2 = 1 − 25/36 = 11/36. (1/3 adds 1/6 + 1/6, which counts the double six twice.)
5 In a class of 30, 18 play tennis, 12 play golf and 5 play both. How many play neither?
Show worked solution
25 play at least one, so 5 play neither. (0 forgets to subtract the 5 who play both.)
6 P(A) = 0.7, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?
Show worked solution
0.7 + 0.4 − 0.2 = 0.9. (1.1 forgets to take away the overlap — and no probability can be above 1.)
7 A student who wears glasses is picked at random. What is the probability that the student is a girl?
Show worked solution
10 wear glasses and 4 are girls: 4/10 = 2/5. (2/7 is P(glasses | girl) — the other way round.)
8 Using the same table, a girl is picked at random. What is the probability that she wears glasses?
Show worked solution
4 of the 14 girls: 4/14 = 2/7. (4/30 is P(girl and glasses), out of the whole class.)
9 The chance of rain is 0.3. If it rains, the chance a bus is late is 0.5; if it is dry, the chance is 0.2. What is the probability that the bus is late?
Show worked solution
0.3 × 0.5 + 0.7 × 0.2 = 0.15 + 0.14 = 0.29. (0.35 adds 0.15 and 0.2 without multiplying the dry branch.)
10 Using the same tree: the bus is late. What is the probability that it rained?
Show worked solution
0.15 ÷ 0.29 = 15/29. (0.5 is P(late | rain) — the reverse condition.)
11 Two fair dice are rolled.
1 P(the total is 7) = 1/6.
2 P(a double) = 1/6.
3 P(the total is 12) = 1/12.
Which statements are correct?
1 P(the total is 7) = 1/6.
2 P(a double) = 1/6.
3 P(the total is 12) = 1/12.
Which statements are correct?
Show worked solution
Six pairs total 7 (1 ✓). Six doubles (2 ✓). Only 6 + 6 totals 12: 1/36, not 1/12 (3 ✗). 1 and 2.
12 A family has three children, and each is equally likely to be a boy or a girl. What is the probability of at least one girl?
Show worked solution
1 − (½)3 = 7/8. (3/8 is P(exactly one girl).)
13 A box holds 4 red and 4 blue pens. Two are taken without replacement. What is the probability of one of each colour?
Show worked solution
2 × (4/8 × 4/7) = 32/56 = 4/7. (2/7 counts only one order.)
14 P(A and B) = 0.18 and P(B) = 0.6. What is P(A | B)?
Show worked solution
0.18 ÷ 0.6 = 0.3. (0.78 adds instead of dividing.)
15 1 If P(A | B) = P(A), knowing that B happened does not change the chance of A.
2 P(A | B) always equals P(B | A).
3 P(not A) = 1 − P(A).
Which statements are correct?
2 P(A | B) always equals P(B | A).
3 P(not A) = 1 − P(A).
Which statements are correct?
Show worked solution
Statement 1 is what P(A | B) = P(A) means (1 ✓). The two conditionals are usually different, as the table showed (2 ✗). A and not A are the only possibilities (3 ✓). 1 and 3.