I am studying for a test and am struggling with this question. Can anyone help?

Suppose you have an unusual deck of cards for a game called Seven-Oh! The probabilities
of drawing each card (1 through 7) in this game are given in the table.

x prob(x)
1 0.11
2 0.08
3 0.23
4 0.22
5 0.05
6 0.15
7 0.16

Suppose someone makes a casino game from these cards. It costs $10 to play, and
you could win $100 if you draw a 5 or you could win $10 if you draw a 2 or a 7.
Otherwise, you win $0. Is this a winning game? Determine the probabilities of
each dollar amount (including $0) and calculate the expected value to answer this
question.

I am studying for a test and am struggling with this question. Can anyone help?

Suppose you have an unusual deck of cards for a game called Seven-Oh! The probabilities
of drawing each card (1 through 7) in this game are given in the table.

x prob(x)
1 0.11
2 0.08
3 0.23
4 0.22
5 0.05
6 0.15
7 0.16

Suppose someone makes a casino game from these cards. It costs $10 to play, and
you could win $100 if you draw a 5 or you could win $10 if you draw a 2 or a 7.
Otherwise, you win $0. Is this a winning game? Determine the probabilities of
each dollar amount (including $0) and calculate the expected value to answer this
question.

Lets generalize this a bit. Instead of winning 100 dollars 5% of the time, we can make it 10 dollars 50% of the time. (For example if you play 100 games, you should win about 5 times, either way you earn 500 dollars.)

The probability of winning 10 dollars back then is 0.5+0.16++0.08

So your probability of winning JUST your money back is 74%, over time, you will end up losing money from not making it back.

Okay, but how do you figure it out by using expected value?

I recall something like this back in high school haha. Something I kind of just thought about.

A winning game would mean you would make back your money, so I just though about the odds of getting $10 back out of 100 games.

Think of it like this:

There's .05 probability of getting the 5. That gets you $100

There's .24 probability of getting the 2 or the 7. (.08 + .16) That gets you $10.

Now, what's the probability of getting nothing? If you get a 1, 3, 4 or 6. What's the probability of getting any of those?

That is basically in the end your three possibilities of what will happen. If you add the probabilities together, then instead of having 7 different cards, you have 3 different possibilities of the outcome.

(You can apply this to all sorts of possibilities, like getting a 5 or higher would be the totals of the 5, 6 & 7, etc.)

Try putting that together and see if you can go from there.

Think of it like this:

There's .05 probability of getting the 5. That gets you $100

There's .24 probability of getting the 2 or the 7. (.08 + .16) That gets you $10.

Now, what's the probability of getting nothing? If you get a 1, 3, 4 or 6. What's the probability of getting any of those?

That is basically in the end your three possibilities of what will happen. If you add the probabilities together, then instead of having 7 different cards, you have 3 different possibilities of the outcome.

(You can apply this to all sorts of possibilities, like getting a 5 or higher would be the totals of the 5, 6 & 7, etc.)

Try putting that together and see if you can go from there.

While this is all correct, the main question is whether this is a winning game or losing game, so finding the probability of drawing the cards you want alone are not enough. Since different cards have different winning values.

I never said it was enough. It was a hint to get the person started. It's the concept of not doing peoples homework for them.