Final answer:
The probability of the topmost card being a diamond is 1/4, while the chance that the bottom card is a number is 9/13. The likelihood of the top two cards sharing the same color is 12/51. The union probabilities depend on whether events overlap and their individual probabilities, and in this case, A₁ and A₂ are not independent.
Step-by-step explanation:
The student asked to determine the probabilities of events A₁, A₂, A₃, and certain unions of these events within the context of a standard deck of 52 playing cards. The standard deck has four suits: clubs, diamonds, hearts, and spades, with each suit having 13 cards ranked from 2 to 10 plus Jack (J), Queen (Q), King (K), and Ace (A). Let's calculate these probabilities step by step.
Event A₁
The probability that the uppermost card on the deck is a diamond (Pr(A₁)) is 13/52 because there are 13 diamonds out of the total 52 cards. Therefore, Pr(A₁) = 1/4.
Event A₂
The probability that the lowermost card on the bottom of the deck is a number card (Pr(A₂)) is 36/52. This is because there are 9 number cards in each suit (2 through 10), totalling 36 number cards in the deck, so Pr(A₂) = 9/13.
Event A₃
The probability that the uppermost two cards on the deck have the same color (Pr(A₃)) requires considering that the first card can be any color. Once a card is drawn, there will be 12 cards of the same color remaining out of the 51 cards left. Thus, Pr(A₃) = 12/51.
Union of Events
For two events, A and B, the probability of their union (Pr(A ⋃ B)) is given by Pr(A) + Pr(B) - Pr(A ∩ B). However, determining whether the events are independent requires checking if Pr(A ∩ B) = Pr(A) × Pr(B). If the equation holds, the events are independent.
Without further steps, we can say that none of these events completely satisfies the criterion for independence. For example, A₁ and A₂ share no common outcome as cards cannot be both at the top and bottom simultaneously, which means they are not independent.