video 6 module 4

Complement of an Event

  • The complement of an event A is denoted as (\text{not } A) or (\overline{A}).

  • It includes all outcomes where event A does not occur.

Example: Skydiving Deaths

  • In a year with 3 million skydiving jumps, 21 resulted in death.

  • To find the probability of not dying while skydiving:

    • Total jumps: 3,000,000

    • Deaths: 21

    • Non-deaths: 3,000,000 - 21 = 2,999,979

  • Probability of not dying: ( \frac{2,999,979}{3,000,000} \approx 0.999993 d) (this probability can be interpreted as a reassuringly high value for potential skydivers).

Identifying Significant Results

  • When observing event probabilities, if the observed outcome significantly deviates from expectations, it suggests that the underlying assumption may be incorrect.

  • Rare Event Rule:

    • A probability threshold for significance is typically (0.05) or 5%.

    • If the probability of an event is less than this threshold, it is considered significant.

Summary of Probability

  • Probability values range from 0 to 1.

  • 0 indicates an impossible event, while 1 indicates a certain event.

  • Probability notation: ( P(A) ) for an event A, and ( P( ext{not } A) ) for its complement.

Odds in Probability

  • Odds against an event are expressed as the ratio of the probability of not A to the probability of A.

  • Odds in favor: Reverse the ratios; probability of event A to the probability of not A.

  • Payoff odds for an event A are the ratio of the net profit to the amount of the bet.

Example: Roulette Odds

  • Betting scenario on number 13 in roulette:

    • Probability of winning: (\frac{1}{38})

    • Probability of not getting 13: (\frac{37}{38})

Part A: Actual Odds Against 13

  • Actual odds against outcome of 13: (\frac{P( ext{not } 13)}{P(13)} = \frac{37/38}{1/38} = 37 : 1)

Part B: Net Profit for Winning Bet on 13

  • Payoff odds provided by the casino: 35 to 1.

  • To calculate net profit:

    • Winning amount: Profit based on the bet, not including the stake.

    • Therefore, profit from a $5 bet: 35(\times) $5 = $175 + original $5 = $180 total return.

Part C: Adjusting Payoff Odds

  • If the casino changed odds to align with actual odds found (37 to 1):

    • With a $5 bet, total return would be $185 ($5(\times)37).

    • Net profit would be $180.

Conclusion of Section 4.1

  • The concepts of complements, significant probabilities, and odds play an important role in understanding basic statistics and probability theory.