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 naming the events
of choosing a red ball, a blue ball, and a green ball as R, B, and G,
respectively. Since the selected ball can only be
one of these three colors, we can conclude that the events are mutually
exclusive. Our aim in this question is to find
the probability that the chosen ball is red or green. This is the probability of the
union of the two events. And we recall for any two mutually
exclusive events 𝑥 and 𝑦, the probability of 𝑥 union 𝑦 is equal to the
probability of 𝑥 plus the probability of 𝑦. This means that we need to find the
sum of the probability that the chosen ball is red and the probability that the
chosen ball is green.
We are told that the probability
that the chosen ball is red is seven times the probability that the chosen ball is
blue. This can be written as shown. We are also told that the
probability that the chosen ball is blue is the same as the probability that the
chosen ball is green.
Finally, since there are only red,
blue, and green balls in the bag and these are mutually exclusive events, we have
the probability of red plus the probability of blue plus the probability of green is
equal to one. Replacing the probability of red
and the probability of green using equations one and two, we have the following
equation. Seven multiplied by the probability
of blue plus the probability of blue plus the probability of blue is equal to
one. This simplifies to nine multiplied
by the probability of blue is equal to one. And the probability that the
selected ball is blue is therefore equal to one-ninth.
This means that the probability
that the chosen ball is green is also equal to one-ninth. And the probability that the chosen
ball is red is seven-ninths. After clearing some space, we can
now calculate the probability that the chosen ball is red or green. This is equal to seven-ninths plus
one-ninth, which in turn is equal to eight-ninths. When a ball is selected from the
bag without looking, the probability that the chosen ball is red or green is
eight-ninths.
