Answer to Question #142026 in Statistics and Probability for Cameron Foster

Question

1. A fair six sided die is rolled followed by a fair coin flip. Please find the following :
(a) Use the multiplication principle to determine the number of outcomes in this experiment (2pts).
(b) Write the sample space below (4pts)
(c) Find the probability of (please leave your solutions as unsimplified fractions) (1pt each)
i. the coin landing tails.
ii. the die rolled a 2
iii. a two is rolled or the coin lands tails
iv. a two is rolled and the coin lands tails.

Answer 1

$\displaystyle (a) \\ \textsf{Number of outcomes of rolling}\\\textsf{a fair die} \, = 6\\ \textsf{Number of outcomes of tossing}\\\textsf{a coin} \, = 2\\ \textsf{By the multiplication principle,}\\ \textsf{the total number of outcomes}\\ \textsf{of rolling a die and tossing}\\\textsf{a coin}\, = 6 \times 2 = 12\\ (b) \\ \textsf{Let}\, H\, \textsf{be the event}\\ \textsf{that a Head was observed and}\\ T \, \textsf{be the event that a Tail}\\ \textsf{was observed.}\\ \textsf{The outcomes of rolling a fair}\\ \textsf{die is}\, 1, 2, 3, 4, 5, 6.\\ \textsf{Thus the total number of possible}\\ \textsf{outcomes are}\\ (H, 1), \, (H, 2), \, (H, 3), \, (H, 4), \\ (H, 5), \, (H, 6), \, (T, 1), \, (T, 2), \\ (T, 3), \, (T, 4), \, (T, 5), \, (T, 6).\\ (c)i.\\ \textsf{The probability that the coin lands}\\ \textsf{a tail is}\, Pr(T) = \frac{1}{2}\\ ii.\\ \textsf{The Probability that die rolled}\\ \textsf{a}\, 2 \, \textsf{is}.\\ Pr(2) = \frac{1}{6}\\ iii. \\ \textsf{The probability that a two}\\ \textsf{is rolled or the coin}\\\textsf{lands a tail is}\\ Pr(T \,\textsf{or}\, 2) = Pr(2) + Pr(T) = \frac{1}{6} + \frac{1}{2} = \frac{4}{6}\\ \textsf{This is because the event}\\ \textsf{is a mutually exclusive event.}\\ iv. \\ \textsf{The probability that a two}\\ \textsf{is rolled and the coin}\\ \textsf{lands a tail is}\\ Pr(T \, \textsf{and}\, 2) = Pr(2) \times Pr(T) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \\ \textsf{This is because the event}\\ \textsf{is an independent event.}$