A bag contains 15 white
balls and a number of black ones. If the probability of drawing a
black ball from the bag is three times that of drawing a white one, find the number
of black balls in the bag.
We can label the number of black
balls in the bag 𝑁 sub 𝐵: that’s what we want to solve for. We’re told the specific number of
white balls in the bag 𝑁 sub 𝑊 is 15. And moreover, we’re told that the
probability of drawing a black ball to that of drawing a white ball — we’ll refer to
it as 𝑃 sub 𝐵 divided by as 𝑃 sub 𝑊 — is equal to three.
Let’s imagine that we’re looking
down into this bag. And the bag has one white ball and
three black balls. If we reach into the bag and pull
out a ball at random, we’ll have a one in four chance of picking up the white ball
and a three in four chance of picking up a black ball. That’s what our picking probability
𝑃 sub 𝐵 over 𝑃 sub 𝑊 equaling three means.
As we consider this example, we see
that the probability ratio is also equal to the ratio of black balls to white
balls. That is, 𝑃 sub 𝐵 over 𝑃 sub 𝑊
is equal to 𝑁 sub 𝐵 over 𝑁 sub 𝑊, the number of black balls to the number of
white balls in the bag. We know that this ratio is equal to
three. And we’re also told in the problem
statement that 𝑁 sub 𝑊 the number of white balls is 15.
So if we take this equation and
multiply both sides of it by 𝑁 sub 𝑊, then we see that on the left-hand side of
this equation that the factor 𝑁 sub 𝑊 cancels out. This leaves us with the result 𝑁
sub 𝐵, the number of black balls, is equal to three times 𝑁 sub 𝑊, the number of
white balls in the bag.
Since 𝑁 sub 𝑊 is 15, we can make
that substitution. And three times 15 is 45. That’s the number of black balls in