Question

Consider a game in which players draw one card at a time from a standard 52-card deck. If a player draws a face card (jack, queen, or king) the player is awarded 16 points. Any other card drawn earns the player 3 points. What is the expected value of drawing a car in the game on the first turn?

@jim_thompson5910

Answer 1

what's the probability of drawing a face card?

Answer 2

16/52

Answer 3

there are 3 face cards (according to their definition and they are: jack, queen, or king)

Answer 4

so it's NOT 4*4 = 16 out of 52

it's actually 4*3 = 12 out of 52 ----> P(face card) = 12/52

Answer 5

the probability of drawing anything else is 40/52

I got 40 because 52 - 12 = 40 (ie there are 40 non-face cards)

Answer 6

Oh okay

Answer 7

so to find the expected value, you do...

Expected Value = E[X]

E[X] = P(Face Card)*V(Face Card) + P(Non Face Card)*V(Non Face Card)

E[X] = (12/52)*(16) + (40/52)*(3)

E[X] = ???



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Notes:

P(Face Card) = probability of drawing a face card
P(Non Face Card) = probability of drawing a non-face card

V(Face Card) = the value (prize or penalty) of getting a nonface card
V(Non Face Card) = the value (prize or penalty) of getting a nonface card

Answer 8

I got 6

Answer 9

Is that right?

Answer 10

sry just noticed these messages

Answer 11

yes it's 6

Answer 12

so on any draw, you expect on average to get roughly 6 points

Answer 13

Can you help me with another one? I tried to solve it but I didnt get the right answer.

Answer 14

alright

Answer 15

Consider a game in which players roll a number cube to determine the number of points earned. If a player rolls a prime number, that many points will be added to the players total. Any other roll will deducted from the players total. What is the expected value of the points earned on a single roll in this game?

Answer 16

it's a cube, so it has 6 sides

Answer 17

I'm assuming the 6 faces are: 1, 2, 3, 4, 5, 6

Answer 18

Of these 6 numbers, the following are prime: 2, 3, 5

Note: 1 is NOT a prime number

Answer 19

Let's go through each roll 1 by 1

Answer 20

okay

Answer 21

Okay

Answer 22

sry let me redo, one sec

Answer 23

Sure thing

Answer 24

Rolling a 1...


P(rolling 1) = 1/6
V(rolling 1) = -1 (you lose 1 point if you roll this)


multiply this out to get (1/6)*(-1) = -1/6


Note: you lose points because 1 is NOT prime



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Rolling a 2...


P(rolling 2) = 1/6
V(rolling 2) = 2 (you earn 2 points if you roll this)


multiply this out to get (1/6)*(2) = 2/6


Note: you earn points because 2 is prime



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Rolling a 3...


P(rolling 3) = 1/6
V(rolling 3) = 3 (you earn 3 points if you roll this)


multiply this out to get (1/6)*(3) = 3/6


Note: you earn points because 3 is prime



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Rolling a 4...


P(rolling 4) = 1/6
V(rolling 4) = -4 (you lose 4 points if you roll this)


multiply this out to get (1/6)*(-4) = -4/6


Note: you lose points because 4 is NOT prime



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Rolling a 5...


P(rolling 5) = 1/6
V(rolling 5) = 5 (you earn 5 points if you roll this)


multiply this out to get (1/6)*(5) = 5/6


Note: you earn points because 5 is prime



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Rolling a 6...


P(rolling 6) = 1/6
V(rolling 6) = -6 (you lose 6 points if you roll this)


multiply this out to get (1/6)*(-6) = -6/6


Note: you lose points because 6 is NOT prime



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Answer 25

Let me know when you want me to continue

Answer 26

You can continue

Answer 27

ok we got these results

-1/6, 2/6, 3/6, -4/6, 5/6, -6/6

Answer 28

Add them up

(-1/6) + (2/6) + (3/6) + (-4/6) + (5/6) + (-6/6)

(-1+2+3+(-4) + 5 + (-6))/6

(-1+2+3-4+5-6)/6

-1/6

Answer 29

So you will expect on average to lose 1/6 of a point every time you roll the die

Answer 30

Oh! Okay thank you!

Answer 31

you're welcome