Expected Value (probability problem)

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SUMMARY

The expected value of a sweepstakes entry can be calculated using the formula E(X) = Σ(X_i * P(X_i)), where X_i represents the prize values and P(X_i) represents their respective probabilities. Participants initially miscalculated the expected value due to a typo in the prize list, specifically mistaking $800,000 for $800. Correcting this error yields an expected value of approximately $1.68 per entry. This discussion highlights the importance of verifying data accuracy in probability calculations.

PREREQUISITES
  • Understanding of expected value calculations in probability theory
  • Familiarity with basic arithmetic operations
  • Knowledge of probability notation and concepts
  • Ability to interpret and analyze data tables
NEXT STEPS
  • Study the concept of expected value in depth, focusing on its applications in gambling and finance
  • Learn about probability distributions and their significance in statistical analysis
  • Explore common pitfalls in probability calculations and how to avoid them
  • Investigate real-world examples of sweepstakes and lottery odds to apply theoretical knowledge
USEFUL FOR

Students studying probability, mathematicians, statisticians, and anyone interested in understanding the mechanics of sweepstakes and expected value calculations.

snoggerT
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Prizes and the chances of winning in a sweepstakes are given in the table below.
Prize: Chances
$25,000,000: 1 chance in 400,000,000
$250,000: 1 chance in 150,000,000
$75,000: 1 chance in 50,000,000
$10,000: 1 chance in 4,000,000
$800,000: 1 chance in 500,000
A watch valued at $70: 1 chance in 8,000

Find the expected value (in dollars) of the amount won by one entry.
E(X) = X_1*P(X_1)+X_2*P(X_2)+...

The Attempt at a Solution



I used the formula listed above and do not get the right answer. I'm really unsure as to why that doesn't work. I use the dollar values for my X and then the chance as my P(X). Can someone explain to me where my fault is in working this problem?
 
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snoggerT said:
Prizes and the chances of winning in a sweepstakes are given in the table below.
Prize: Chances
$25,000,000: 1 chance in 400,000,000
$250,000: 1 chance in 150,000,000
$75,000: 1 chance in 50,000,000
$10,000: 1 chance in 4,000,000
$800,000: 1 chance in 500,000
A watch valued at $70: 1 chance in 8,000

Find the expected value (in dollars) of the amount won by one entry.



E(X) = X_1*P(X_1)+X_2*P(X_2)+...



The Attempt at a Solution



I used the formula listed above and do not get the right answer. I'm really unsure as to why that doesn't work. I use the dollar values for my X and then the chance as my P(X). Can someone explain to me where my fault is in working this problem?
Please show us your work. It might be that you made an arithmetic mistake that is causing your answer to be incorrect.

As a side note, it's odd that the probability of winning $25M is relatively high, compared to the probability of winning a prize that is only 1% of that.

Also, you show 1 chance in 4 million of winning $10,000, for 1 chance in 1/2 million of winning $800,000. Are you sure that you have written all the prizes and their probabilities exactly as presented in the problem?
 
Mark44 said:
Please show us your work. It might be that you made an arithmetic mistake that is causing your answer to be incorrect.

As a side note, it's odd that the probability of winning $25M is relatively high, compared to the probability of winning a prize that is only 1% of that.

Also, you show 1 chance in 4 million of winning $10,000, for 1 chance in 1/2 million of winning $800,000. Are you sure that you have written all the prizes and their probabilities exactly as presented in the problem?

- I copied the values straight from the webwork homework. I do find it very odd that the numbers go from $10,000 @ 1/4,000,000 chance to $800,000 @ 1/500,000 chance. But here is how I worked the problem:

E(X) = (25,000,000*(1/400,000,000))+(250,000*(1/150,000,000))+(75,000*(1/50,000,000))+(10,000*(1/4,000,000))+(800,000*(1/500,000))+(70*(1/8,000)) = 1.67691666666667
 
snoggerT said:
E(X) = (25,000,000*(1/400,000,000))+(250,000*(1/150,000,000))+(75,000*(1/50,000,000))+(10,000*(1/4,000,000))+(800,000*(1/500,000))+(70*(1/8,000)) = 1.67691666666667
I don't see anything wrong with it. I get the same answer.
 
Mark44 said:
I don't see anything wrong with it. I get the same answer.

- yeah, I'm at a loss. Maybe the numbers are showing up wrong in relation to the actual answer.
 
Alright, I figured it out. The $800,000 value is supposed to be $800. So I suppose my professor made a typo when setting the problem up.
 
snoggerT said:
Alright, I figured it out. The $800,000 value is supposed to be $800. So I suppose my professor made a typo when setting the problem up.
If you get the "right" answer when you replace $800K by $800, that probably is what happened.
 

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