Question

A standard 52 card deck is being used in an exciting experiment! A card is drawn randomly from the deck, its

information is recorded, then the card is returned to the deck and it is thoroughly shuffled.

(a) Determine the probability that if we perform this process 6 times, we get exactly 3 diamonds, and exactly 1 spade.

(b) If we repeat this experiment 11 times, what is the probability that we get three times as many clubs as hearts. (c) Determine the probability that if this process is repeated 5 times, there will be exactly 2 diamonds drawn,

and at most 1 jack.
(d) Suppose we repeat this experiment 7 times. Determine the probability that we draw exactly 3 clubs and

exactly 1 heart, if we know that we drew exactly 2 red cards.

Answer 1

Solution:-

Given that

A standard 52 card deck is being used in an exciting experiment! A card is drawn randomly from the deck, its information is recorded, then the card is returned to the deck and it is thoroughly shuffled.

(a) Determine the probability that if we perform this process 6 times, we get exactly 3 diamonds, and exactly 1 spade.

Since the experiment is done with replacement.

we can get 6 cards in ways

P(getting 3 diamonds and 1 spade) =

{Here is the number of ways of getting 2 cards that are not diamond or spade}

P(geting 3 diamonds and 1 spade) =

= 0.000976

(b) If we repeat this experiment 11 times, what is the probability that we get three times as many clubs as hearts.

We can get 11 cards in ways

H : heart c : clubs

Favourable cases = {1H 3C ; 2H, 6C}

P(getting 3 times many clubs as hearts)

= 0.00392

P(Getting 3 times as much clubs as hearts) = 0.00392

(c) Determine the probability that if this process is repeated 5 times, there will be exactly 2 diamonds drawn, and at most 1 jack.

We can get 5 cards in ways

D : diamonds J : jack

Favorable cases = {2D 0J; 2D 1J}

P(getting 2D and atmost 1 J)  

  

  

= 0.0117

P(getting 2 diamonds and atmost 1 jack) = 0.0117
(d) Suppose we repeat this experiment 7 times. Determine the probability that we draw exactly 3 clubs and exactly 1 heart, if we know that we drew exactly 2 red cards.

We can get 7 cards in ways

c : club H: heart

Since we have 2 red cards and want only 1 heart, that mean other red card must be a diamond.

P(getting exactly 3 clubs and 1 heart)

Since there are 5 black cards, that means 2 are spade as we only want 3 clubs.

P(getting exactly 3 clubs and 1 heart)

= 0.000061

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