Question

A​ scratch-off lottery ticket has 14 concealed spaces among which are 4 symbols saying​ "Win!". The person who bought the ticket scratches the black yucky stuff off 3​ spaces, winning an instant​ $10 if all three are winners.​ What's the probability of winning this​ game? I'm confused about this please help!

Answer 1

Exact answer as a fraction = 1/91

Approximate answer in decimal form = 0.010989

Approximate answer in percent form = 1.0989%

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Explanation:

Let's think of the problem another way.

Let's say that we have 14 ping pong balls labeled 1 through 14. Furthermore, lets set up the first four balls {1,2,3,4} to be labeled with a star sticker. The rest don't have a star. The star stickers indicate a winning ball. All of the ping pong balls are then put into a bag where we will randomly pull out 3 of them. Replacements will not be made (ie the ball won't be put back). The goal is to get 3 star stickers in a row.

One possible combination is {1,2,7} which is close but not a winner since that 7 is there. Another possible combination is {5,8,12} which is not even close to a winner (since we didn't get 1,2,3 or 4).

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We have two questions we need to answer:

1. How many possible combinations are there?
2. How many combinations are a winner?

We'll use the nCr combination formula

`_nC_r = (n!)/(r!*(n-r)!)`

We use the combination formula, instead of the permutation formula, because order does not matter. Something like {1,2,7} is the same as {7,1,2}.

If you plug in n = 14 and r = 3, you should find that 14 C 3 = 364.

I'll list the steps in the attached image below (to save space here).

This means there are 364 different ways to play the game. In other words, there are 364 ways to pull out three ping pong balls.

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That wraps up question 1. Now onto question 2.

We have 4 winning balls and 3 slots to fill, so we will plug n = 4 and r = 3 into the combination formula. I'll skip the steps because they're fairly similar to earlier. You could also use Pascal's Triangle to help. Whichever method you use, you should find that 4 C 3 = 4.

note: we have 4 winning balls and 3 slots to fill, so you can think of it like "how many ways are there to not choose a ball?" and that answer would be 4.

So this means there are 4 ways to win the game.

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In summary so far, we found

1. There are 364 ways to play the game
2. There are 4 ways to win the game

This means the probability of winning is 4/364 = 1/91

Use your calculator to find that 1/91 = 0.010989 approximately which converts over to the approximate percentage of 1.0989%

Answer 2

Explanation:

since this has something to do with chance im guessing maybe percentage of winnign this

so

4/14 x 100 so 28.57% probability of winning this game i believe