To find the possible values for the probability that Anji goes to the cinema on a particular Friday, we will use the given probabilities and work through the problem step-by-step.
Let's define the following probabilities:
- [tex]\( P(A) \)[/tex]: Probability that Anji goes to the cinema on a particular Friday.
- [tex]\( P(K) \)[/tex]: Probability that Katrina goes to the cinema on a particular Friday.
We are given:
1. The probability that both Anji and Katrina go to the cinema: [tex]\( P(A \cap K) = \frac{1}{3} \)[/tex].
2. The probability that at least one of them goes to the cinema: [tex]\( P(A \cup K) = \frac{5}{6} \)[/tex].
Using the formula for the probability of the union of two events, we have:
[tex]\[
P(A \cup K) = P(A) + P(K) - P(A \cap K)
\][/tex]
Substituting the given values:
[tex]\[
\frac{5}{6} = P(A) + P(K) - \frac{1}{3}
\][/tex]
We can simplify this equation to:
[tex]\[
\frac{5}{6} + \frac{1}{3} = P(A) + P(K)
\][/tex]
[tex]\[
\frac{5}{6} + \frac{2}{6} = P(A) + P(K)
\][/tex]
[tex]\[
\frac{7}{6} = P(A) + P(K)
\][/tex]
Now we have two key equations:
1. [tex]\( P(A) \cdot P(K) = \frac{1}{3} \)[/tex]
2. [tex]\( P(A) + P(K) = \frac{7}{6} \)[/tex]
We can solve this system of equations to find possible values for [tex]\( P(A) \)[/tex] and [tex]\( P(K) \)[/tex].
Solving the second equation for [tex]\( P(K) \)[/tex]:
[tex]\[
P(K) = \frac{7}{6} - P(A)
\][/tex]
Substitute this into the first equation:
[tex]\[
P(A) \cdot \left( \frac{7}{6} - P(A) \right) = \frac{1}{3}
\][/tex]
[tex]\[
P(A) \cdot \frac{7}{6} - P(A)^2 = \frac{1}{3}
\][/tex]
[tex]\[
\frac{7}{6}P(A) - P(A)^2 = \frac{1}{3}
\][/tex]
To eliminate the fraction, multiply through by 6:
[tex]\[
7P(A) - 6P(A)^2 = 2
\][/tex]
Rearrange:
[tex]\[
-6P(A)^2 + 7P(A) - 2 = 0
\][/tex]
This is a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = -6 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -2 \)[/tex]. We can solve this quadratic equation using the quadratic formula:
[tex]\[
P(A) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the values:
[tex]\[
P(A) = \frac{-7 \pm \sqrt{7^2 - 4(-6)(-2)}}{2(-6)}
\][/tex]
[tex]\[
P(A) = \frac{-7 \pm \sqrt{49 - 48}}{-12}
\][/tex]
[tex]\[
P(A) = \frac{-7 \pm \sqrt{1}}{-12}
\][/tex]
[tex]\[
P(A) = \frac{-7 \pm 1}{-12}
\][/tex]
This gives us two solutions:
[tex]\[
P(A) = \frac{-7 + 1}{-12} = \frac{-6}{-12} = \frac{1}{2}
\][/tex]
[tex]\[
P(A) = \frac{-7 - 1}{-12} = \frac{-8}{-12} = \frac{2}{3}
\][/tex]
Thus, the possible values for the probability that Anji goes to the cinema on a particular Friday are:
[tex]\[
P(A) = \frac{1}{2} \quad \text{or} \quad P(A) = \frac{2}{3}
\][/tex]
