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Question Video: Determining the Probability of Union of Two Events Mathematics

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 will begin by letting 𝑅, 𝐡, and 𝐺 be the events of selecting a red, blue, and green ball, respectively. We are told that the probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue. This can be written as shown: 𝑃 of 𝑅 is equal to seven multiplied by 𝑃 of 𝐡. We are also told that the probability the chosen ball is blue is the same as the probability that the chosen ball is green. 𝑃 of 𝐡 is therefore equal to 𝑃 of 𝐺.

We are asked to find the probability that the chosen ball is red or green. However, before doing this, we will calculate the probabilities of selecting each of the three color balls. Since probabilities sum to one, we know that the probability of selecting a red ball plus the probability of selecting a blue ball plus the probability of selecting a green ball must equal one. Replacing 𝑃 of 𝑅 with seven 𝑃 of 𝐡 and 𝑃 of 𝐺 with 𝑃 of 𝐡, we have the following equation. This simplifies to nine 𝑃 of 𝐡 is equal to one. And dividing through by nine, we see that the probability of selecting a blue ball is one-ninth. This means that the probability of selecting a green ball is also equal to one-ninth. And since the probability of selecting a red ball is seven times this, this is equal to seven-ninths.

We are now in a position where we can find the probability that the chosen ball is red or green. And since the two events are mutually exclusive, we can use the fact that the probability of 𝐴 union 𝐡 is equal to the probability of 𝐴 plus the probability of 𝐡, where the probability of 𝐴 union 𝐡 is the probability that either one of the events occurs.

We simply need to add the probability that the chosen ball is red to the probability the chosen ball is green. Since the denominators are the same, we simply add the numerators. And we can therefore conclude that the probability that the chosen ball is red or green is eight-ninths. It is also worth noting that we could’ve calculated this by subtracting the probability that the ball is blue from one. One minus one-ninth is also equal to eight-ninths.

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