Suleika has six cards numbered 1 to 6.
(a) She takes one card at random, records the number and replaces the card.
(i) Write down the probability that the number is 5 or 6.
(ii) Suleika does this 300 times.
Find how many times she expects the number 5 or 6.
(b) Suleika takes two cards at random, without replacement.
(i) Find the probability that the sum of the numbers on the two cards is 5.
(ii) Find the probability that at least one of the numbers on the cards is a square number.
(a)(i) Ans: \(\frac{1}{3}\)
There are 2 favorable outcomes (5 or 6) out of 6 possible cards. Thus, the probability is \(\frac{2}{6} = \frac{1}{3}\).
(a)(ii) Ans: 100
Multiply the probability \(\frac{1}{3}\) by the number of trials (300): \(\frac{1}{3} \times 300 = 100\).
(b)(i) Ans: \(\frac{2}{15}\)
Possible pairs summing to 5: (1,4), (2,3), (4,1), (3,2). There are 4 favorable outcomes out of \(6 \times 5 = 30\) total outcomes, giving \(\frac{4}{30} = \frac{2}{15}\).
(b)(ii) Ans: \(\frac{3}{5}\)
Square numbers in 1-6: 1, 4. Probability of neither card being square is \(\frac{4}{6} \times \frac{3}{5} = \frac{2}{5}\). Thus, the probability of at least one square is \(1 – \frac{2}{5} = \frac{3}{5}\).
