Two cards are drawn in succession from a standard deck of 52 cards without replacing the first one before the second.

a) Are these outcomes are independent? Why?

b) Find P ( 3 on 1st card and 10 on the 2nd card)

c) Find P( 10 on the 1st card and 3 on 2nd card)

d) Find the Probability of drawing 10 and a 3 in either order

a) The choice of the first card without replacement changes the probability of the second card.

b) 4/52 and 4/51. To find probability that all events would occur, multiply the probability of the individual events.

c) Same

d) "either-or" --> add individual probabilities. Add b and c.

a) To determine if the outcomes are independent, we need to check if the probability of the second event is affected by the outcome of the first event. In this case, since the first card is not replaced before drawing the second card, the outcomes are dependent. The probability of drawing a certain card for the second event will be affected by which card was drawn in the first event.

b) To find the probability of getting a 3 on the first card and a 10 on the second card, we need to multiply the probability of drawing a 3 on the first card (4/52, since there are 4 cards with the value of 3 in a deck of 52 cards) by the probability of drawing a 10 on the second card given that a 3 was already drawn (4/51, since there are 4 cards with the value of 10 left in the remaining 51 cards).

So, P(3 on 1st card and 10 on 2nd card) = (4/52) * (4/51) = 16/2652 = 4/663

c) To find the probability of getting a 10 on the first card and a 3 on the second card, we need to multiply the probability of drawing a 10 on the first card (4/52) by the probability of drawing a 3 on the second card given that a 10 was already drawn (4/51).

So, P(10 on 1st card and 3 on 2nd card) = (4/52) * (4/51) = 16/2652 = 4/663

d) To find the probability of drawing a 10 and a 3 in either order, we can add the probabilities found in parts (b) and (c) since these represent both possible orders of drawing the cards.

So, P(drawing 10 and 3 in either order) = P(3 on 1st card and 10 on 2nd card) + P(10 on 1st card and 3 on 2nd card)
= (4/52) * (4/51) + (4/52) * (4/51)
= 16/2652 + 16/2652
= 32/2652
= 8/663

Therefore, the probability of drawing a 10 and a 3 in either order is 8/663.