Please show work.. A set of 6 cards are numbered from 1 to 6 1 point) A set of 6

Question

Please show work.. A set of 6 cards are numbered from 1 to 6

1 point) A set of 6 cards are numbered from 1 to 6. Two cards are drawn successively with replacement. Let E be the event "the second card has a (strictly) larger number than the first card." Find P(E) Hint: Let i be the number on the first card you draw and A be the event "the first card drawn has the number i on it." What is the probability of event A,? Then what is the probability of drawing a card with a number strictly larger than i on the second draw? For this problem, try counting how many cards have a number larger than i in the deck, and remember that the cards are replaced after the first draw, so the second draw still comes from 6 cards. Then the probability of drawing card i on the first card, and then drawing a card with a number strictly larger than i on the second draw, is the product of the two probabilities given in the previous paragraph. We will see this in more detail in Chapter 3. To see how this hint helps, determine what the event En Ai (or Ai n E) describes.

Explanation / Answer

Total number of possibilities for two cards = 6*6 = 36

Now, if first card value is i:

i = 1: Second card can be from 2 to 6 so 5 possibilities for second card

i = 2: Second card can be from 3 to 6 so 4 possibilities for second card

i = 3: Second card can be from 4 to 6 so 3 possibilities for second card

i = 4: Second card can be from 5 to 6 so 2 possibilities for second card

i = 5: Second card can be 6 so 1 possibility for second card

i = 6: Second card can't take any value because 6 is larget so 0 possibilities for second card

Hence,

Number of outcomes in which second card is larger than first = 5 + 4 + 3 + 2 + 1 + 0 = 15

Therefore,

P(E) = 15/36 = 5/12

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