you draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck.
a) are the outcomes on the two cards independent? why?
b) find P(3 on 1st card and 10 on 2nd)
c) find P(10 on 1st card and 3 on 2nd)
d) find the probability of drawing a 10 and a 3 in either order.
8/663
a) No, the outcomes on the two cards are not independent. The reason is that after drawing the first card, reshuffling the deck will change the composition of the remaining cards and can affect the probability of drawing specific cards on the second draw.
b) To find the probability of drawing a 3 on the first card and a 10 on the second card, we need to multiply the individual probabilities of drawing each card. The probability of drawing a 3 is 4/52 (since there are 4 cards with the value 3 out of 52 total cards), and after reshuffling, the probability of drawing a 10 is also 4/52. Therefore, P(3 on 1st card and 10 on 2nd) = (4/52) * (4/52) = 16/2704.
c) Similarly, to find the probability of drawing a 10 on the first card and a 3 on the second card, we still need to multiply the individual probabilities of drawing each card. After reshuffling, the probability of drawing a 10 is 4/52, and the probability of drawing a 3 is also 4/52. Therefore, P(10 on 1st card and 3 on 2nd) = (4/52) * (4/52) = 16/2704.
d) To find the probability of drawing a 10 and a 3 in either order, we need to consider two scenarios:
1) Drawing a 10 on the first card and a 3 on the second card: Probability = P(10 on 1st and 3 on 2nd) = 16/2704.
2) Drawing a 3 on the first card and a 10 on the second card: Probability = P(3 on 1st and 10 on 2nd) = 16/2704.
Since these two scenarios are mutually exclusive (either one or the other can occur), we can add their probabilities together to get the overall probability: (16/2704) + (16/2704) = 32/2704 = 1/84.
a) In this scenario, the outcomes on the two cards are independent. This is because before drawing the second card, the first card is put back and the deck is reshuffled. This means that each card has an equal chance of being drawn on the second draw, regardless of what was drawn on the first draw.
b) To find the probability of getting a 3 on the first card and a 10 on the second, we first need to determine the probability of each individual event.
There are 4 cards with a value of 3 in a standard deck of 52 cards. Therefore, the probability of drawing a 3 on the first card is 4/52.
After putting the first card back and reshuffling, there are 4 cards with a value of 10 in the deck. So, the probability of drawing a 10 on the second card is also 4/52.
To find the probability of both events happening, we multiply the probabilities together: (4/52) * (4/52) = 1/169.
Therefore, the probability of drawing a 3 on the first card and a 10 on the second card is 1/169.
c) Similarly, to find the probability of getting a 10 on the first card and a 3 on the second, we use the same reasoning. The probability of drawing a 10 on the first card is 4/52, and the probability of drawing a 3 on the second card is also 4/52.
So, the probability of drawing a 10 on the first card and a 3 on the second card is (4/52) * (4/52) = 1/169.
d) To find the probability of drawing a 10 and a 3 in either order, we add the probabilities of both scenarios from parts b) and c).
P(10 on 1st card and 3 on 2nd) = 1/169
P(3 on 1st card and 10 on 2nd) = 1/169
Adding these probabilities together:
P(drawing a 10 and a 3 in either order) = 1/169 + 1/169 = 2/169.
Therefore, the probability of drawing a 10 and a 3 in either order is 2/169.
a) Yes, because of replacement, the probabilities do not change for the second card.
b, c) probability of both events occurring is found by multiplying the probability of the individual events.
d) probability of either-or events occurring is found by adding the probability of the individual events.