Video: Determining the Probability of Union of Two Events

A bag contains red, blue, and green balls, and one is to be selected without looking. The probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue. The probability that the chosen ball is blue is the same as the probability that the chosen ball is green. Find the probability that the chosen ball is red or green.

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Video Transcript

A bag contains red, blue, and green balls, and one is to be selected without looking. The probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue. The probability that the chosen ball is blue is the same as the probability that the chosen ball is green. Find the probability that the chosen ball is red or green.

We want to find the probability that we select a red or a green ball when we draw one out. We know that these events are mutually exclusive because the ball could not be red and green at the same time. The probability of the ball being both red and green is zero. And that means the probability that the ball is red or green will be equal to the probability that the ball is red plus the probability that the ball is green.

So now we need to find those values. We know that the probability that the ball is red is equal to seven times the probability that the ball is blue. And the probability that the ball is green is equal to the probability that the ball is blue. We can also say that the probability of red plus the probability of green plus the probability of blue has to equal one.

Since the sum of the probabilities of all possible outcomes is always equal to one. In this equation, if we substitute the probability of blue and for the probability of green, we can substitute seven times the probability of blue and for red. So seven times the probability of blue plus the probability of blue plus the probability of blue has to equal one.

We can combine like terms and say that we have nine times the probability of blue is equal to one. And that means we could divide both sides of this by nine to show the probability of selecting blue is equal to one-ninth. If the probability of selecting blue is one-ninth, then the probability of selecting green is also one-ninth as they are equal. The probability of selecting red is equal to seven times the probability of selecting blue, which means it will be seven-ninths. To find the probability of red and green, then we combine seven-ninths and one-ninth to get eight-ninths.

It’s probably worth noting here that once we found the probability of blue, we could’ve found the probability that it is not blue. Because there’re only red, green, and blue in the bag, the probability that it is not blue will be the same as the probability of it being red or green, which again would be eight-ninths.

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