Question

The probability that Color Me Dandy wins a horse race at Batavia Downs given good track conditions is $0.60$. The probability of good track conditions on any given day is $0.85$. What is the probability that Color Me Dandy wins or the track conditions are good?

(a)$0.94$

(b)$0.51$

(c)$0.49$

(d) $0.06$

(e) The answer cannot be determined from the given information.

Short answer

The probability that Color Me Dandy wins or the track conditions are good is:

(e) The answer cannot be determined from the given information.

Step-by-step solution

Definition

General multiplication rule:

$$\begin{gathered} P\left(A\text{and}B\right)=P(A\cap B) \\ =P\left(A\right)\times P(B\mid A) \\ =P\left(B\right)\times P(A\mid B) \end{gathered}$$

General addition rule for any two events:

$P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right)$

Explanation

$P(\text{ColorMeDandywins}\mid Goodtrack)=0.60$

$P(Goodtrack)=0.85$

By the general multiplication rule, we can then determine P (Color Me Dandy wins and Good track) (which is the product of P( Color Me Dandy wins \mid Good track ) and P( Good track ).

However, we note that using the general addition rule, we need $P$(Color Me Dandy wins) to determine $P$(Color Me Dandy wins or Good track), and so we can't predict the answer from the given data because we don't know the probability $P$. (Color Me Dandy wins)..