Betting odds are usually stated against the event happening...

Betting odds are usually stated against the event happening (against winning). The odds against event $W$ are the ratio $\frac{P(\text {not} W)}{P(W)}=\frac{P\left(W^{*}\right)}{P(W)}$ In horse racing, the betting odds are based on the probability that the horse does not win. (a) Show that if we are given the odds against an event $W$ as $a: b,$ the probability of not $W$ is $P\left(W^{\circ}\right)=\frac{a}{a+b} .$ Hint: Solve the cquation $\frac{a}{b}=\frac{P(W)}{1-P\left(W^{\circ}\right)}$ for $P\left(W^{c}\right)$ (b) In a recent Kentucky Derby, the betting odds for the favorite horse, Point Given, were 9 to $5 .$ Use these odds to compute the probability that Point Given would lose the race. What is the probability that Point Given would win the race? (c) In the same race, the betting odds for the horse Monarchos were 6 to 1 . Use these odds to estimate the probability that Monarchos would lose the race. What is the probability that Monarchos would win the race? (d) Invisible Ink was a long shot, with betting odds of 30 to $1 .$ Use these odds to estimate the probability that Invisible Ink would lose the race. What is the probability the horse would win the race? For further information on the Kentucky Derby, visit the web site of the Kentucky Derby.

Betting odds are usually stated against the event happening (against winning).
The odds against event $W$ are the ratio $\frac{P(\text {not} W)}{P(W)}=\frac{P\left(W^{*}\right)}{P(W)}$
In horse racing, the betting odds are based on the probability that the horse does not win.
(a) Show that if we are given the odds against an event $W$ as $a: b,$ the probability of not $W$ is $P\left(W^{\circ}\right)=\frac{a}{a+b} .$ Hint: Solve the cquation $\frac{a}{b}=\frac{P(W)}{1-P\left(W^{\circ}\right)}$ for $P\left(W^{c}\right)$
(b) In a recent Kentucky Derby, the betting odds for the favorite horse, Point Given, were 9 to $5 .$ Use these odds to compute the probability that Point Given would lose the race. What is the probability that Point Given would win the race?
(c) In the same race, the betting odds for the horse Monarchos were 6 to 1 . Use these odds to estimate the probability that Monarchos would lose the race. What is the probability that Monarchos would win the race?
(d) Invisible Ink was a long shot, with betting odds of 30 to $1 .$ Use these odds to estimate the probability that Invisible Ink would lose the race. What is the probability the horse would win the race? For further information on the Kentucky Derby, visit the web site of the Kentucky Derby.

Key Concepts

Betting Odds

Betting odds are a numerical representation that compares the likelihood of an event occurring to its failure. They are typically expressed as a ratio (e.g., a:b), where the numbers indicate the relative chances of winning versus losing, and are used to both inform bettors and set potential payouts.

Conversion of Odds to Probability

Converting odds to probability involves relating the given ratio to the total number of outcomes (winning and losing). If the odds against an event are given as a:b, the probability of the event not occurring is calculated as a divided by (a + b), and the probability of the event occurring is then b divided by (a + b). This conversion is critical in translating betting information into standard probabilistic measures.

Complementary Probability

The concept of complementary probability refers to the relationship between an event and its complement, where the probability of an event occurring plus the probability of it not occurring equals one. This principle is fundamental when solving for the unknown probability in problems where odds against or for an event are provided.

Transcript

00:01 This question is based on probability and odds, the concept of the odds.

00:07 Okay, let us look at part a.

00:10 Let us look at part a.

00:14 All right.

00:15 What are they saying? they are saying show that if we are given the odds against an event w as a is to b.

00:23 If we are given the odds against the event w against w against w, against w, the odds are a is to b.

00:42 Okay the probability of not w is a upon a plus b the probability of not w is a upon a plus b the probability of not w is a upon a plus b okay we are already given in the question that the the odds against the event w are the ratio of p not w of p not w upon pw these are the odds against right and this is already given to us as a by p right all right let us just look at this from a conceptual point of me let's say that w is the event of me winning a game there are in all a plus b matches okay out of those i lose a matches and i win b matches right so what will be the odds against me what are the odds against me? the odds against me are a is to b.

01:59 What is the probability that i am losing? i have lost a matches, right? so let me just do this in red.

02:08 I have lost a matches and i am winning b matches.

02:12 So what is the probability that i have lost? i have lost a.

02:15 And what is the total number of matches? a plus b.

02:18 So what is the probability of me losing? the probability of me losing will be a upon a plus b right this is what this conceptually means and me losing means me not winning and what was not winning i said let me say that w is me winning right so me not winning is what a upon a plus b and i think this is what we wanted to prove right p w not is equal to a upon a plus b this is what we wanted to prove so conceptually we have proved this now let us move on to part b all right part b says in a recent kentucky derby, the betting odds for the favorite horse, point given were 9 to 5.

03:10 So in the favor for point given, in favor of point given, it was 9 to 5.

03:17 It was 9 to 5.

03:21 Okay.

03:22 Meaning if there were 14 games, he would win 9, there is a probability that he will win 9 games and lose 5.

03:30 Now what is the question the question says what use these odds to compute the probability that point given would lose the race okay lose the race means he's losing in 5 out of 14 so what is the probability that he would lose the race it would be 5 by 14 okay then they are saying what is the probability that point given would win the race okay so what is the probability that point given would win the race he is winning 9 times out of 14 so the probability in that case would be 9 by 14, right? okay.

04:11 Now moving on to part c.

04:18 In the same race, the betting odds for the horse monarchos were 6 to 1...

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