Write the sample space and calculate the probability based on the given information.
Two dice are rolled. What is the probability that the sum is a prime number greater than 5?
Sample Space: Rolling two dice gives n(S) = 36. Event: Sum is a prime number greater than 5 (7 or 11). Favourable Outcomes: Sum of 7 occurs in 6 ways, sum of 11 occurs in 2 ways. n(E) = 8. Probability: P(E) = n(E)/n(S) = 8/36 = 2/9.
A bag contains 4 red, 3 green, and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are of different colours?
Sample Space: Total balls = 9, drawing 2 gives n(S) = 36. Event: Both balls are of different colours. Favourable Outcomes: P(different) = 1 - P(same colour). P(both red) = 12/72, P(both green) = 6/72, P(both blue) = 2/72. P(same) = 20/72 = 5/18. Probability: P(E) = 1 - 5/18 = 13/18.
Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?
Sample Space: Tossing three coins gives n(S) = 8. Event: First coin shows heads and exactly two heads occur. Favourable Outcomes: The outcomes are HHT and HTH. n(E) = 2. Probability: P(E) = n(E)/n(S) = 2/8 = 1/4.
A four-digit number is formed using the digits 1, 2, 3, and 4 with no repetition. What is the probability that the number is even?
Sample Space: Forming a 4-digit number from 1,2,3,4 without repetition gives n(S) = 24. Event: The number is even (ends in 2 or 4). Favourable Outcomes: If it ends in 2 (6 ways) or 4 (6 ways), n(E) = 12. Probability: P(E) = n(E)/n(S) = 12/24 = 1/2.
A student takes a multiple-choice test with 3 questions, each having 4 options (A, B, C, D), with only one correct answer. What is the probability that the student guesses and gets exactly 2 answers correct?
Sample Space: 3 questions with 4 options each gives n(S) = 64. Event: Exactly 2 answers correct. Favourable Outcomes: 3 ways to choose 2 correct questions × 3 wrong options for the third = 9. Probability: P(E) = n(E)/n(S) = 9/64.
Explanation
The solutions apply the fundamental concepts of probability from the textbook: defining the sample space (S) to find the total possible outcomes n(S), identifying the event (E) to find the favourable outcomes n(E), and calculating probability as P(E) = n(E)/n(S). For part (i), we identify primes > 5. For part (ii), we use complementary probability for efficiency. For part (iii), we restrict outcomes based on the first coin. For part (iv), we use permutations for even numbers. For part (v), we use combinations and the counting principle for independent choices.