Question

A deck of 10 cards consists of 3 hearts cards (♥) cards, 3 diamonds (♦) cards, and 4 spades (♠) cards. The deck is shuffled until the cards are in random order, and then it is determined (without any reshuffling) whether the following three gamblers have won their bets. Alice bets that the 3 hearts cards are together (next to each other, in any order) in the deck, Bob bets that the 3 diamonds cards are together in the deck, and Carol bets that the 4 spades cards are together in the deck. If Alice wins her bet, she gets $5, if Bob wins his bet he also gets $5, while Carol gets $20 if she wins her bet. Nobody wins or loses money on a lost bet.

(a) Compute the probability that Alice wins her bet, the probability that Bob wins his bet, and the probability that Carol wins her bet. Give the three answers as reduced fractions.

(b) Compute the probability that all three gamblers win their bets.

(c) Compute the expected combined dollar amount that the three gamblers get in this game. Give the answer as a reduced fraction.

Answer 1

ANSWER::

(a)

10 cards can be arranged in 10P10 = 10! ways

Keeping 3 hearts cards as one unit, Number of ways 3 hearts together = Number of ways to arrange 8 cards together (7 remaining cards + 1 unit of 3 hearts cards) = 3! 8! (Three heart cards as one unit can be arranged in 3! ways)

Keeping 3 diamonds cards as one unit, Number of ways 3 diamonds together = Number of ways to arrange 8 cards together (7 remaining cards + 1 unit of 3 diamonds cards) = 3! 8! (Three diamonds cards as one unit can be arranged in 3! ways)

Keeping 4 spades cards as one unit, Number of ways 4 spades together = Number of ways to arrange 7 cards together (6 remaining cards + 1 unit of 4 spades cards) = 4! 7!   (Four spades cards as one unit can be arranged in 4! ways)

Probability that Alice wins the bet = Number of ways 3 hearts together / Total number of ways 10 cards can be arranged

= 3! * 8! / 10! = 1/15

Probability that Bob wins the bet = Number of ways 3 diamonds together / Total number of ways 10 cards can be arranged

= 3! * 8! / 10! = 1/15

Probability that Carol wins the bet = Number of ways 4 spades together / Total number of ways 10 cards can be arranged

= 4! * 7! / 10! = 1/30

(b)

Keeping 3 hearts, 3 diamonds and 4 spades cards as three different units, Number of ways 3 hearts, 3 diamonds and 4 spades cards together = Number of ways to arrange 3 units

= 3! (3! * 3! * 4!) / 10! = 1/700   (3 hearts, 3 diamonds and 4 spades cards can be arranged in 3! * 3! * 4! ways)

(c)

If any of the two wins the bet, the third person will also win the bet. So, the probability that to get combined dollar mount of 5+5+20 = 30 is 1/700

Probability that only Alice wins the bet = 1/15 - 1/700 = (140 - 3)/ 2100 = 137 / 2100. In this case, the winning amount is $5

Probability that only Bob wins the bet = 1/15 - 1/700 = (140 - 3)/ 2100 = 137 / 2100. In this case, the winning amount is $5

Probability that only Carol wins the bet = 1/30 - 1/700 = (70 - 3)/ 2100 = 67 / 2100. In this case, the winning amount is $20

Expected combined dollar amount = 30 * (1/700) + 5 * (137 / 2100) + 5 * (137 / 2100) + 20 * (67 / 2100)

= (30 * 3 + 5 * 137 + 5 * 137 + 20 * 67) / 2100

= 2800 / 2100

= 4/3

(OR) TRY THIS

There are 10 cards

So, number of possible combinations of 10 cards = 10!

But out of 10 cards 3 are Hearts, 3 are diamond , 4 are spade.

So, 3 cards ,3 cards and 4 cards are similar in themselves.

Now, the number of ways if arrangement of 10 cards if 3 are hearts, 3 are diamonds and 4 are spades are

.....(1)

Now, Alice wins her bet if all the hearts are together.

So, if all the hearts are together then 3 hearts cards can be treated as one card.

Now, here we can say the number of cards are (1 +3 +4) 8 out of which 3 diamonds, 4 spades cards are same.

Now, the number of ways of arranging if 3 hearts cards are together are

....(2)

Similarly

Now, Bob wins his bet if all the diamonds are together.

So, if all the diamonds are together then 3 diamonds cards can be treated as one card.

Now, here we can say the number of cards are (3+1 +4) 8 out of which 3 hearts, 4 spades cards are same.

Now, the number of ways of arranging if 3 diamonds cards are together are

...(3)

Again Similarly

Now, Carol wins her bet if all the spades are together.

So, if all the spades cards are together then 3 spades cards can be treated as one card.

Now, here we can say the number of cards are (3 +3 +1) 7 out of which 3 hearts and 3 diamonds cards are same.

Now, the number of ways of arranging if 3 spades are together are

...(4)

If all three win their bets then

all 3 heart cards are together and can be treated as one card ,

all diamond cards are together and can be treated as one card ,

all spades are together and can be treated as one card

So, number of ways of arranging the cards of all 3 win their bets is

....(5)

Now,

(a)

The probability that Alice wins her bet is

On putting the values from equations (1) and (2), we get

Similarly

The probability that Bob wins his bet is

On putting the values from equations (1) and (3), we get

Similarly

The probability that Carol wins her bet is

On putting the values from equations (1) and (4), we get

Hence, probabilities that Alice win her bet, Bob win his bet and Carol win her bet are 1/15 , 1/15 and 1/30 respectively.

(b)

The probability that all the gamblers wins their bets is

On putting the values from equations (1) and (5), we get

Hence, probability that all the three gamblers win their bet is 1/700.

(c)

Since the probabilities of winning if Alice , Bob and Carol are P(A)=1/15, P(B)=1/15 and P(C) =1/30 respectively.

And On winner the his/her bet Alice , Bob and Carl's receives A= $ 5, B=$ 5 , C =$ 20 dollars respectively.

So, the expected amount of combined money win by the gamblers

E(x) = A×P(A) + B×P(B) + C×P(C)

On putting the values, we get

E(Amount) = [5×1/15] + [5×1/15] + [25×1/30]

E(Amount) = 1/3 + 1/3 + 5/6

E(Amount)= 2/6 + 2/6 + 5/6

E(amount) = 9/6 = 3/2 dollars.

Hence, expected combine amount win by all the gamblers is $ 3/2

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