189 Odds of winning a horse race. Handicappers for horse races express their beliefs about the probability of each horse winning a race in terms of odds . If the probability of event E is P(E) , then the odds in favor of E are P(E)

to 1 - P(E). Thus, if a handicapper assesses a probability of .25 that Smarty Jones will win the Belmont Stakes, the odds in favor of Smarty Jones are 25100 to 75100 , or 1 to 3. It follows that the odds against E are 1 - P(E) to P(E) , or 3 to 1 against a win by Smarty Jones. In general, if the odds in favor of event E are a to b , then P(E) = a>(a + b).

a. A second handicapper assesses the probability of a win by Smarty Jones to be 13. According to the second handicapper, what are the odds in favor of a Smarty Jones win?

b. A third handicapper assesses the odds in favor of Smarty Jones to be 1 to 1. According to the third handicapper, what is the probability of a Smarty Jones win?

c. A fourth handicapper assesses the odds against Smarty Jones winning to be 3 to 2. Find this handicapper’s assessment of the probability that Smarty Jones will win.