# Fundamentals of probability part II

02/20/2020

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## Outline

• The following topics will be covered in this lecture:
• Odds
• Compound events
• Conditional probability
• The multiplication rule

## Odds

• Expressions of likelihood are often given as odds, such as $$50:1$$ (or “50 to 1”).

• Because the use of odds makes many calculations difficult, statisticians, mathematicians, and scientists prefer to use probabilities.

• The advantage of odds is that they make it easier to deal with money transfers associated with gambling, so they tend to be used in casinos, lotteries, and racetracks.

• Note – in the three definitions that follow, the actual odds against and the actual odds in favor are calculated with the actual likelihood of some event;

• however, the payoff odds describe the relationship between the bet and the amount of the payoff.
• The actual odds correspond to actual probabilities of outcomes, but the payoff odds are set by racetrack and casino operators.

• Racetracks and casinos are in business to make a profit, so the payoff odds will not be the same as the actual odds.

### Odds continued

• Actual odds against event $$A$$ – this is the probability of event $$\overline{A}$$ relative to the event $$A$$, i.e.,

$\frac{P\left(\overline{A}\right)}{P(A)}$

• Actualy odds against is usually expressed in the form of $$a:b$$ (or “$$a$$ to $$b$$”), where $$a$$ and $$b$$ are integers having no common factors.
• Actual odds in favor of event $$A$$ – this is the probability of event $$A$$ relative to the event $$\overline{A}$$, i.e.,

$\frac{P(A)}{P\left(\overline{A}\right)}$

• If the odds against $$A$$ are $$a:b$$, then the odds in favor of $$A$$ are $$b:a$$.
• Payoff odds against event $$A$$ – this is the ratio of net profit (if you win) to the amount bet:

$\text{payoff odds against event }A = (\text{net profit}):(\text{amount bet})$

### Odds example

• If you bet $$5$$ dollars on the number $$13$$ in roulette, your probability of winning is $$\frac{1}{38}$$ and the payoff odds are given by the casino as $$35:1$$.

• Discuss with a neighbor: what are the actual odds for and the actual odds against winning with a bet on $$13$$?

• Let's note that if $$A=$$"winning with a bet on $$13$$", we can write $$P(A)=\frac{1}{38}$$.
• Therefore, the probability of not winning is $P\left(\overline{A}\right) = 1 - \frac{1}{38} = \frac{37}{38}$
• If the actual odds for a bet on $$13$$ are thus given as, $\frac{P(A)}{P\left(\overline{A}\right)} = \frac{\frac{1}{38}}{\frac{37}{38}} = \frac{1}{37},$ or as odds, $$1:37$$.
• If we have the actual odds for a bet on $$13$$ as $$1:37$$ then the actual odds against are given as $$37:1$$.
• Recall our formula for payoff odds, $\text{payoff odds against event }A = (\text{net profit}):(\text{amount bet})$

• Discuss with a neighbor: how much net profit would you make if you win by betting on $$13$$?

• If we bet one dollar, we net a profit of $$35$$ dollars, so we can multiply this ratio to obtain $$175:5$$ as the net profit to bet.
• The net profit is $$175$$ which means the casino gives you your winnings of $$175$$ plus $$5$$ for your original bet.

### Odds example continued

• If you bet $$5$$ dollars on the number $$13$$ in roulette, your probability of winning is $$\frac{1}{38}$$ and the payoff odds are given by the casino as $$35:1$$.

• Discuss with a neighbor: if the casino was not operating for profit and the payoff odds were changed to match the actual odds against $$13$$, how much would you win with a bet of $$5$$ dollars if the outcome were $$13$$?

• If the payoff odds were equal to the actual odds against, we would be computing, $37:1 = (\text{Net profit}):(\text{ammount bet}).$
• Thus if we used this rule, we could multiply the ratio by five again to find $$185:5$$.
• Our net profit would therefore be $$185$$ dollars on a $$5$$ dollar bet – this means the casino would owe you $$185$$ plus $$5$$ dollars for your original bet.

## Compound events

• We will often be concerned not with one event $$A$$ but some combination of some event $$A$$ and some event $$B$$.
• Compound event – formally we define a compound event as any event combining two or more simple events.
• There are two key operations joining events
1. “OR” – in mathematics we refer to “or” as a non-exclusive “or”.
• The meaning of this for “$$A$$ or $$B$$” is – event $$A$$ occurs, event $$B$$ occurs, or both events $$A$$ and $$B$$ occur.
• We will not consider the exclusive “or”, i.e. either event $$A$$ occurs, or event $$B$$ occurs, but not both.
2. “AND” – in mathematics we refer to “and” in an exclusive sense.
• The meaning of this for “$$A$$ and $$B$$” is – both event $$A$$ and event $$B$$ occurs.
• The operations “and” and “or” join events together in a way that we can compute the probability of the joint events.
• We will develop some tools describing how to compute probabilities of these compound events from the individual probabilities.
• A key concept is how we compute the probability of events without double counting the ways they can occur. Courtesy of Bin im Garten CC via Wikimedia Commons

• Suppose we want to compute the probability of two events $$A$$ and $$B$$ joined by the compound operation “or”.
• We read the statement, $P(A \text{ or } B)$ as he probability of event:
• $$A$$ occuring,
• event $$B$$ occuring, or
• both $$A$$ and $$B$$ ocurring.
• Intuitively, we can express the probability in terms of all the ways $$A$$ can occur and all the ways $$B$$ can occur, if we don’t double count.
• Let all the ways that $$A$$ can occur be represented by the red circle to the left.
• Let all the ways that $$B$$ can occur be represented by the dashed circle to the left.
• Discuss with a neighbor: suppose we count all the ways $$A$$ can occur and all the ways $$B$$ can occur.
• If we take the sum of the total of all ways $$A$$ occurs and the total of all ways $$B$$ occurs, does this give the total of all ways $$A$$ or $$B$$ occurs?
• Consider, if there is an overlap where both $$A$$ and $$B$$ occur simultaneously, then summing the total of all ways $$A$$ occurs and the total of all ways $$B$$ occurs double counts the the cases where both $$A$$ and $$B$$ occur. Courtesy of Bin im Garten CC via Wikimedia Commons

• Let us consider the statement ${\color{red} {P(A)}} + P(B),$ is equal to the sum of the total of all ways that $$A$$ occurs and the total of all ways that $$B$$ occurs, relative to all possible outcomes.
• Discuss with a neighbor: what term “$$\cdot$$” is needed in $$P\left( \cdot \right)$$ below to eliminate the double counting? $P(A\text{ or }B)= {\color{red} {P(A)}} + P(B) - P\left( \cdot \right)$
• We count the cases where $$A$$ and $$B$$ both occur twice, as these cases are included in both $${\color{red} {P(A)}}$$ and $$P(B)$$.
• Therefore, the addition rule for compound events is given as, $P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B)$
• Discuss with a neighbor: notice that if $$P(A\text{ and } B) = 0$$ then $P(A\text{ or }B) = P(A) + P(B),$ is an accurate statement.
• If $$P(A\text{ and } B) = 0$$, what does this say about the relationship between $$A$$ and $$B$$?
• This says that events $$A$$ and $$B$$ never occur simultaneously.
• An easy example is for $$A=$$"coin flip lands heads" and $$B=$$"coin flip lands tails" – these two events never occur simultaneously and $$P(A\text{ or } B)= P(A) + P(B)$$.
• Two events that never occur sumultaneously are called disjoint or mutually exclusive events, corresponding to when there is no overlap.