WASSCE SHS
A probability is a fraction from 0 to 1. Without replacement, the second draw sees a smaller total than the first.
If outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the number of possible outcomes. The probability of the complementary event is 1 minus that value. Probabilities of all the mutually exclusive outcomes that cover every possibility add to 1. Multiply probabilities when you want both of two independent events. Add them when you want one or the other of two mutually exclusive events, and be careful not to add when the events overlap.
Without replacement the draws are not independent. A bag with 4 red, 3 blue and 1 green has 8 beads. The probability of red first is 4/8. If that bead stays out, 3 red remain and 7 beads remain, so the probability of a second red is 3/7. The probability of both is the product, 4/8 × 3/7 = 3/14. Say whether the question replaced the bead. The fraction changes.
Simplify the fraction if it cancels cleanly. 12/56 should become 3/14. A decimal is accepted only if it is exact or the question asked for one. "Without replacement" must change the second denominator. A candidate who writes 4/8 × 4/8 has replaced the bead and answered a different question. Show the product. The final fraction alone can lose the method mark.
These are practice questions in the style of the paper, with a model answer and a mark scheme. They are not a promise of what your paper will ask.
3 marks
A bag contains 4 red beads, 3 blue beads and 1 green bead.
(a) Find the probability that a bead taken at random is blue. (1 mark)
(b) Find the probability that it is not red. (2 marks)
(a) 3/8.
(b) Beads that are not red: 4 out of 8, so 4/8 = 1/2. Alternatively 1 − 4/8 = 1/2.
3 marks
Two beads are taken one after another without replacement. Find the probability that both are red. (3 marks)
(4/8) × (3/7) = 12/56 = 3/14.
1 mark
How would the second fraction change if the first bead had been replaced? (1 mark)
It would stay 4/8, because the bag would be back to 4 red out of 8.
More questions on this topic are inside GLearn, marked as you answer, with the reason written out. The first ones need no account.
Practise Elective Mathematics freeThe totals for the next draw fall by one bead, and by one of the colour you removed. With 4 red out of 8, the chance both are red is 4/8 times 3/7, which is 3/14. If you replace the bead, the second chance is still 4/8.
Subtract the probability of the event from 1. The probability of not red, when 4 of 8 beads are red, is 1 − 4/8, which is 1/2. The probabilities of red and not red add to 1. All WASSCE Elective Mathematics past questions