For the spinner below, assume that the pointer can never lie on a border line. Find the following probabilities. (Enter the probabilities as fractions.)
(a) $P(A)$
(b) $P(B)$
(c) $P(C)$
We can use the same process to find $P(A)$ and $P(B)$ if we know the values of the other two probabilities.
Step 1 :Let's denote the probability of the spinner landing on $A$ as $P(A)$, on $B$ as $P(B)$, and on $C$ as $P(C)$.
Step 2 :Since the spinner is guaranteed to land on exactly one of the three regions, we know that the sum of the probabilities of it landing in each region will be 1. So we have the equation $1 = P(A) + P(B) + P(C)$.
Step 3 :If we know the values of $P(A)$ and $P(B)$, we can substitute them into the equation to find $P(C)$.
Step 4 :For example, if $P(A) = \frac{1}{3}$ and $P(B) = \frac{5}{12}$, we can substitute these values into the equation to get $1 = \frac{1}{3} + \frac{5}{12} + P(C)$.
Step 5 :Solving this equation for $P(C)$, we get $P(C) = 1 - \frac{1}{3} - \frac{5}{12}$.
Step 6 :Calculating the right side of the equation, we get $P(C) = \frac{1}{4}$.
Step 7 :So the probability of the spinner landing on $C$ is $\boxed{\frac{1}{4}}$.
Step 8 :We can use the same process to find $P(A)$ and $P(B)$ if we know the values of the other two probabilities.