Lesson 1B: Chance Odds

The chance odds of an event E are closely related to chance probability of E, but they are not equal. In this lesson we learn what chance odds are and their relation to chance probabilities.

Deﬁnition 1: Odds in Favor and Against

Let Ω = {ω1, ··· , ωn} be an equally-likely sample space and E = {ωi1 , ··· , ωik } ⊂ Ω be an event in Ω. Let n(E), n(Ec), and n(Ω) represent the number of sample points in E, Ec, and Ω, respectively (i.e., the cardinality of each set). Then,

a) Odds in Favor: Odds in favor of the occurrence of E is deﬁned as

n(E) k the number of ways that E can occur OF (E) := = = . n(Ec) n − k the number of ways that E cannot occur Example: Draw a card from a deck of cards. Let, E be the event that we draw a face card.

n(E) 12 • OF (E) = n(Ec) = 40 . n(E) 12 • P (E) = n(Ω) = 52 . b) Odds Against: Odds against the occurrence of E is deﬁned as

n(Ec) n − k the number of ways that E cannot occur OA(E) := = = . n(E k the number of ways that E can occur Example: Draw a card from a deck of cards. Let, E be the event that we draw a face card.

n(Ec) 40 • OA(E) = n(E) = 12 . n(E) 12 • P (E) = n(Ω) = 52 .

Chance Odds and Chance Probabilities

• Recall that if E is an event in an equally-likely sample space, then n(E) P (E) = . n(Ω)

• Since E and Ec are mutually-exclusive and their union is Ω, it follows that

n(Ω) = n(E) + n(Ec).

• Thus, if we add the numerator and the denominator of OF (E) or OA(E), we obtain n(Ω) (in equivalent reduced fraction value). We can then convert the OF (E) or OA(E) to P (E). • Example 1: Roll a pair of dice once and record the sum of the numbers that face up. Let E be the event that the sum of the numbers that face up is 7. Then, E = {(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)}. Thus, 6 6 6 OF (E) = , which implies that P (E) = = . 30 6 + 30 36 8 • Example 2: If OF (E) = 5 , what is P (E)? 8 8 P (E) = = . 8 + 5 13

17 • Example 3: If OA(E) = 11 , what is P (E)? 11 11 P (E) = = . 11 + 17 28

• Similarly, if we subtract the numerator from the denominator of P (E), we obtain n(Ec) (in equivalent reduced fraction value). We can then convert the P (E) to OF (E) or OA(E).

23 • Example 1: If P (E) = 31 , then 23 23 OF (E) = = . 31 − 23 8

31 − 23 8 OA(E) = = . 23 23

7 • Example 2: If P (E) = 23 , then 7 7 OF (E) = = . 23 − 7 16

23 − 7 16 OA(E) = = . 7 7

Definition 2: Payoff Odds (odds offered by casinos)

The odds that casinos offer refer to the amount of money a gambler can win based on the amount of money they bet. Let, E be an event on which a gambler intends to place a bet. Then, the payoff odds may be offered in two ways.

a) Payoff Odds in Favor of E: • Let’s say a casino offers a 2 : 1 payoff odds in favor of E. • This means that if you win, for every $2 that you bet, you will get your $2 back plus $1 winnings from the casino. • In this case your stake is $2: YS = 2, casino’s stake is $1: CS = 1, and the total stake is $3: TS = 3. • Thus, payoff odds in favor of E means: YS Payoff Odds in Favor of E = POF (E) = . CS

b) Payoff Odds Against E: • Let’s say a casino offers a 2 : 1 payoff odds against E. • This means that if you win, for every $1 that you bet, you will get your $1 back plus $2 winnings from the casino. • In this case your stake is $1: YS = 1, casino’s stake is $2: CS = 2, and the total stake is $3: TS = 3. • Thus, payoff odds against E means: CS Payoff Odds Against E = POA(E) = . YS

Remarks:

Clearly, the questions that come to mind are: (i) What is the relationship between payoﬀ odds and chance odds?; (ii) How do you know that a game you are playing is a fair game? The answers to these questions are provided by the Cardano’s Rule.

Cardano’s Rule:

If we roll a die, we are as likely to roll an even number as an odd number (OF (even) = OF (odd)). Thus, if the die is fair, then the bets are set according to this equality; And if the die is not fair, then the bets are made larger or smaller in proportion to the departure from the true equality.

Remarks:

• According to Cardano’s Rule, if a game is fair, the payoﬀ odds in favor of E are equal to the the chance odds in favor of E. Namely,

(payoﬀ odds in favor of E) = (chance odds in favor of E).

Thus, we get the expression YS n(E) POF (E) = OF (E) ⇐⇒ = . CS n(Ec)

• Next, in order to determine when a game is fair ﬁrst recall the following identities:

i) Identities for P (E), P (Ec), and POF (E): n(E) P (E) = n(Ω) n(Ec) P (Ec) = = 1 − P (E) n(Ω) YS YS POF (E) = = CS TS − YS ii) Thus, if a game is fair, it must be true that: YS n(E) = , by def. CS n(Ec) YS n(E)/n(Ω) = TS − YS n(Ec)/n(Ω) YS P (E) = , in view of n(Ec)/n(Ω) = P (Ec) = 1 − P (E) TS − YS 1 − P (E)

YS(1 − P (E)) = P (E)(TS − YS)

YS − YS · P (E) = P (E) · TS − P (E) · YS

YS = TS · P (E) Hence, a game is fair iﬀ the last identity is satisﬁed.

Example: A roulette table has a rectangular shape and consists of 36 small congruent squares numbered 1 through 36, and two small rectangles numbered 0 and 00. Half of the small squares are red and the other half are black. Both small rectangles are green. Casinos provide many ways to play and bet on a roulette table.

A casino oﬀers even payoﬀ odds on red. Namely, POF (red) = 1 : 1. Is this a fair bet?

• A bet is fair if YS = TS · P (E).

• Even payoﬀ odds means: YS = 1 and CS = 1. Thus, TS = 2.

18 • P (red) = 38 . • Thus, 18 36 TS · P (E) = 2 · = = $0.95 6= $1 = Y S. 38 38 • Hence, this is not a fair bet. In order for this game to be fair, YS should be $0.95. • The $0.05 that the casino keeps is called a house percentage.