One urn contains two black balls (labelled B1 and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W1 and W2). One of the two urns is chosen at random. Next a ball is randomly drawn from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball.
(a) Write the sample space showing all possible outcomes.
(b) What is the probability that two black balls are chosen?
(c) What is the probability that two balls of opposite colour are chosen?
(a) \( S = \{ B_1B_2, B_1W, B_2B_1, B_2W, WB_1, WB_2, W_1B, W_1W_2, W_2B, W_2W_1 \} \)
(b) \( \dfrac{1}{6} \)
(c) \( \dfrac{2}{3} \)
A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that
(a) all three balls are white
(b) all three balls are red
(c) one ball is red and two balls are white
(a) \( \dfrac{5}{143} \)
(b) \( \dfrac{28}{143} \)
(c) \( \dfrac{40}{143} \)
If the letters of the word ASSASSINATION are arranged at random, find the probability that
(a) four S’s come consecutively in the word
(b) two I’s and two N’s come together
(c) all A’s are not coming together
(d) no two A’s are coming together
(a) \( \dfrac{2}{143} \)
(b) \( \dfrac{2}{143} \)
(c) \( \dfrac{25}{26} \)
(d) \( \dfrac{15}{26} \)
A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.
\( \dfrac{7}{13} \)
A sample space consists of 9 elementary outcomes \( e_1, e_2, ..., e_9 \) whose probabilities are:
P(e₁) = P(e₂) = .08, P(e₃) = P(e₄) = P(e₅) = .1, P(e₆) = P(e₇) = .2, P(e₈) = P(e₉) = .07.
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₇, e₉}.
(a) Calculate P(A), P(B), and P(A ∩ B)
(b) Using the addition law of probability, calculate P(A ∪ B)
(c) List the composition of A ∪ B, and calculate P(A ∪ B) by adding probabilities.
(d) Calculate P( B̄ ) from P(B), also calculate P( B̄ ) directly from elementary outcomes.
(a) \( P(A) = 0.25 \), \( P(B) = 0.32 \), \( P(A \cap B) = 0.17 \)
(b) \( P(A \cup B) = 0.40 \)
(c) \( P(A \cup B) = 0.40 \)
(d) \( P(\overline{B}) = 0.68 \)
Determine the probability p, for each of the following events:
(a) An odd number appears in a single toss of a fair die.
(b) At least one head appears in two tosses of a fair coin.
(c) A king, 9 of hearts, or 3 of spades appears in drawing a card from a well-shuffled deck.
(d) The sum of 6 appears in a single toss of a pair of dice.
(a) \( \dfrac{1}{2} \)
(b) \( \dfrac{3}{4} \)
(c) \( \dfrac{3}{26} \)
(d) \( \dfrac{5}{36} \)