4.(4 + 3 marks) In an ordinary deck of 52 playing cards, if a person draws...

4.(4 + 3 marks) In an ordinary deck of 52 playing cards, if a person draws a king or a jack, he is paid $10; if he draws a queen, he is paid $5.If he draws any other card, he will pay $4. a. If X is the gain for the person who plays the game, construct a probability distribution for X. b. Find the expected gain?

Solutions

Expert Solution

Here we start by listing out the chances of gettingthe particular cards. Then we calculate their respective probability. This probability will be same for getting or losing the money since it is dependent on the card drawn. Here 'X' is the payout

P(card drawn) = No. of the type of card / Total cards.

Card	No.	Prob (No. / 52)
King	4	0.0769
Jack	4	0.0769
Queen	4	0.0769
Other cards	40	0.7692

Other cards are remaining out of 52. They not 4 king, queens or jacks. Therefore 52-4-4-4 = 40

Card	X	P(X=x)	X*P(X=x)
King or Jack	10	0.1538	1.5385
Queen	5	0.0769	0.3846
Other	-4	0.7692	-3.0769
Total		1	-1.1538

Here we combine the probabiltiy of king and jack since they both can get $10.

This is a pmf because each 0 <p<1, the sum of all is '1' and each event is distinct.

Expected Gain = X * P(X =x)

Expected gain = -$1.15

That is this will get an expected loss of $1.15.

orchestra answered 1 year ago

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