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Prize Winner, Part 1

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Dependent event

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Therefore, to summarize that last exercise, we know:

    • probability of popping a pink balloon given a pink balloon is popped equals eleven nineteenths
    • probability of popping a green balloon given a pink balloon is popped equals four nineteenths
    • probability of popping a blue balloon given a pink balloon is popped equals three nineteenths
    • probability of popping a red balloon given a pink balloon is popped equals one-nineteenth

This is a picture of the balloon game. There are five columns of balloons, and four rows of balloons. There are twelve pink balloons, four green balloons, three blue balloons, and one red balloon.

In order to win a prize, a gamer must pop a pink balloon first. Thus, the second step is dependent on the first. The events are dependent. This means The probability of A and B does not equal the probability of A times the probability of B and therefore the probability of B given A equals the probability of A and B divided by the probability of A, which does not equal the product of the probability of A and the probability of B divided by the probability of A which does not equal the probability of B. . The events A and B, that is popping a pink and then a green, are dependent on each other.

It may seem that the games that Khalid, Justyce, Andrew, Marissa and Allyson are developing have a relatively high probability of winning. For example, if we want to win a small prize, we have to pop a green balloon after popping a pink one. From above, we know that probability of popping a green balloon given a pink balloon is popped equals four nineteenths, which is a 21% chance.

Be careful to interpret conditional probabilities correctly. This 21% chance is the chance of winning after a pink balloon is popped. This is not the probability of winning overall.

To find the probability of winning, follow these steps:

  • The probability of a pink balloon being popped first is 1220 .
  • Then, the probability of popping a green balloon given the pink balloon is popped first  is 419 .
  • The probability of both of these events happening is (1220)(419)=0.126 .

There is a 12.6% chance of winning a small prize.

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