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
    • The addition rule
    • 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.

Addition rule

Venn diagram of events \( A \) and \( B \) with nontrivial intersection.

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.

Addition rule continued

Venn diagram of events \( A \) and \( B \) with nontrivial intersection.

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.

Addition rule continued