Theory question 106
Probability(a) The table shows the number of limes and apples of the same size in a bag.
Good: Limes 10, Apples 8
Bad: Limes 6, Apples 6
If two of the fruits are picked at random, one at a time, without replacement, find the probability that: (i) both are good limes; (ii) both are bad fruits; (iii) one is a good apple and the other a bad lime
(b) Solve the equation log3(4x + 1) − log3(3x − 5) = 2
Show the model answer
Total fruits = 30.
(a)(i) P = 10/30 × 9/29 = 3/29.
(ii) There are 12 bad fruits: P = 12/30 × 11/29 = 22/145.
(iii) Good apple then bad lime or bad lime then good apple: 2 × (8/30 × 6/29) = 16/145.
(b) log3[(4x + 1)/(3x − 5)] = 2, so (4x + 1)/(3x − 5) = 9.
4x + 1 = 27x − 45, 23x = 46, x = 2 (check: 3x − 5 = 1 > 0).
Without replacement, the second fraction has one fewer in the denominator (29).