SOLUTION: HOMEWORK HELP PLEASE I DO NOT UNDERSTAND A person selects a card from an ordinary deck of cards. If it is a black card, she wins $2. If it is a red card between or including 3

Question 892913: HOMEWORK HELP PLEASE I DO NOT UNDERSTAND
A person selects a card from an ordinary deck of cards. If it is a black card, she wins $2. If it is a red card between or including 3 and 7, she wins $10. If it is a red face card, she wins $25 and if it is a black jack, she wins an additional $100. If it is any other card, she wins nothing. Find the expectation of the game.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the expected value is equal to the sum of (the probability of an outcome * the expected value of that outcome).

your problem states:

if the card is black, she wins 2 dollars.
if it is red between and including 3 and 7, she wins 10 dollars.
if it is a ref face card, she wins 25 dollars.
if it is a black jack, she wins an additional 100 dollars.
if it is any other card, she wins nothing.
she has not bet any money of her own so she can't lose.
she just might not win.

the probability of getting a black card is 26/52.
she wins 2 dollars if she gets a black card.
her expected earnings are 26/52 * 2 = 52/52 dollars.

the probability of getting a red card with a face value of 3 through 7 is equal to 10/52.
she wins 10 dollars if she gets a red card between and including 3 and 7.
her expected earnings are 10/52 * 10 = 100/52 dollars.

the probability of getting a red face card is 6/52.
she wins 25 dollars if she gets a red face card.
her expected earnings are 6/52 * 25 = 150/52 dollars.

the probability of getting a black jack is 2/52.
she wins 100 dollars if she gets a black jack.
her expected earnings are 2/52 * 100 = 200/52 dollars.

note that, if she gets a black jack, she has won a total of 102 dollars.
100 dollars because it's a black jack and 2 dollars because it's a black card.
that's what they mean by an additional 100 dollars.

if she gets anything else she gets nothing so they don't count.

her total expected value is:

52/52 + 100/52 + 150/52 + 200/52 = 502/52 = 9.653846154 dollars.

that equals 9.65 rounded to 2 decimal places.

there are 52 cards in the deck.
there are 26 black cards in the deck (13 spades and 13 clubs).
2 of the black cards are black jacks (1 is a spade and 1 is a club).
there are 26 red cards in the deck (13 hearts and 13 diamonds).
10 of the red cards have a face value of 3 through 7 (5 are hearts and 5 are diamonds).
6 of the red cards are face cards (3 are hearts and 3 are diamonds).
a face card is a jack or a queen or a king.
the cards that are in the deck that will not earn any money are:
ace of hearts and ace of diamonds.
10 of hearts and 10 of diamonds.
9 of hearts and 9 of diamonds.
8 of hearts and 8 of diamonds.
2 of hearts and 2 of diamonds.
those total up to 10 cards in the deck that will earn nothing.
if you add up the cards in the deck that will earn money, that total will be 42.
26 of them are black.
6 of them are red face cards.
10 of them are red cards with a face value of 3 through 7.
add those up and you get 42 cards that earn money.
the 2 black jacks that earn an additional 100 dollars are included in the count of 26 black cards.

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