In business (lotteries and casinos included), the owner wants to ensure that the business makes a profit. So that expected value (average profit) is always more than the expected loss (customer wins/payout). Why would this be? This week we learn about the expected value (mean) of a probability distribution. This would apply to casinos, horse or car races, bets on athletic events, etc.
Google your state’s (Virginia) Powerball or megamillions and calculate the probability of winning the jackpot. Most sites report odds instead of probability, so you’ll need to do a simple conversion.
Share a link that takes us to the odds information on your state’s Powerball or megamillions.
In your original post, answer the following:
- Create a probability table for this distribution
- What is your expected gain or loss from playing?
- If you played 100 times how much would you expect to win or lose?
- After looking at this information and doing a few calculations, would you play? Why or why not?
- Is it binomial? I’ll leave this for you to investigate and explain.
In your responses to your classmates, address the following:
- Would you play? Discuss in-depth.
- How could the binomial distribution described in the original post be adjusted to favor the ‘house’ more? How could it be adjusted to favor the player more?”
Convert the odds to a probability
To convert from odds to a probability, divide the odds by one plus the odds. So, to convert odds of 1/9
to a probability, divide 1/9 by 10/9 to obtain the probability of 0.10.
Example:
Taking the odds of 1:9 and knowing I’ll win 1/10 times I have a probability of .10 of winning.
This means I have a probability of .90 (the complement) of losing. Let’s say if I spend a dollar, I could
possibly win 2 dollars. If I lose, I lose the dollar.
My distribution looks like this.
X P(x)
-$1 .9 (lose)
$2 .1 (win)
Calculate expected value: E(x) = Sum (x* P(x))
E(x) = -1(.9) + 2(.1) = -.7 I expect to lose 70 cents ON AVERAGE each time I play.
If I play 100 times, I’ll lose .7(100) — expected loss in one game times the number of times I play
I expect to lose $70 dollars if I play 100 times. Why do I not expect to lose 100 dollars?