Logical Reasoning

PT112 · S3 · Q21 Sandy: I play the Bigbucks

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Sandy: I play the Bigbucks lottery—that’s the one where you pick five numbers and all the players who have picked the five numbers drawn at the end of the week share the money pot.

Sandy's Conclusion

It's best to play Bigbucks only after there have been a few weeks with no winners.

Sandy's Evidence (because)

The money pot increases each week that there is no winner.

Alex's Conclusion

You're more likely to win the lottery when the money pot is small.

Alex's Evidence (because)

That's when the fewest other people are playing.

Evaluate

What a psycho question task: read two different arguments and find a flaw that shows up in one of them?

Okay, let's analyze both reasoning moves:

Given that ... the money pot increases each week that there is no winner, Are we convinced that ... you should only play Bigbucks after a few weeks of no winners?

No, not necessarily. Maybe the tickets become more expensive when the money pot gets bigger. Maybe so many extra people start buying tickets when the jackpot gets high that you're more likely to have multiple winners who have to split the jackpot. (i.e. better to win a smaller jackpot by yourself than have to split a bigger jackpot among multiple winners)

Given that ... the fewest people play when the money pot is small Are we convinced that ... you're more likely to win when the money pot is small?

No, we don't have to accept that. If you're involved in a raffle, then the fewer people involved (or more precisely the fewer tickets purchased) the better your odds. In a raffle, there's a guaranteed winner because one of the purchased ticket numbers will be randomly drawn.

So if there are 80 tickets, you have a 1 in 80 chance of winning. If there are 800 tickets, you have a 1 in 800 chance of winning.

Meanwhile, Bigbucks is not a raffle. You pick five numbers, and the lottery randomly picks five numbers, and if they all match then you win (at least a share) of the money pot.

The probability of winning is determined here by the odds that the five digit number you chose will match the randomly drawn number they choose.

If each of the five numbers are just one digit (from 0 - 9), then there are 105 possible numbers. You thus have a 1 in 100,000 chance of choosing the winning five digit number. It doesn't matter how many people are playing the lottery. Your odds are still 1 in 100,000 of your number matching the winning number. (the addition of other players only increases the odds that you might have to split the pot if you win, but it doesn't change your odds of winning)

Goal

Alex's flaw seems like the more egregious, more nameable flaw, so we can probably expect to see an answer saying that .

21.

Which one of the following most accurately describes a mistake in the reasoning of one of the two speakers?

  1. Sandy holds that the chances

    Not True

    Sandy doesn't make any comments about how playing multiple times does / doesn't affect your odds of winning. She's only talking about when it's worth playing, not how many times someone plays. Everyone's common sense knows that the more you play, the more chances you have to win, so for someone to hold that your chances of winning are unaffected by how many times you play would have a really crazy sense of probability.

    2% picked this

  2. Correct

    Alex holds that the chances

    Why this is right

    This is certainly a true description of Alex's argument. His holding is the conclusion "you're more likely to win", which is about the chances of Sandy's winning. His evidence is about how many people are playing when the pot is large vs. small. If we didn't diagnose why this was flawed, we might guess this answer simply because it's at least an accurate description. As discussed above, this is flawed thinking because your odds of winning Bigbucks are based purely on the math of how many possible ways there are to pick five numbers. If there are ten options for each number (0 - 9), then there are 100,000 ways to pick five numbers, and thus the odds of any five number ticket winning are 1 out of 100,000, regardless of how many people play.

    Skill tested: Flaw · how this choice captures the argument's function is the move to repeat next time.

    64% picked this

  3. Sandy holds that the chances

    Bad Conclusion Match

    Sandy never talks about the chances of anyone winning. Her conclusion is about when it's optimal to play based on the size of the pot. She isn't ever claiming that your chances of winning change. She's implying that the potential payout of winning is bigger in some cases than others.

    13% picked this

  4. Alex holds that the chances

    Opposite

    Actually, because of Alex's flawed belief that the number of people playing affects one's chances of winning, he would actually think that your chances of winning are affected by whether someone won the week before. He would think, "If someone won last week, then this week's money pot is very small, so the fewest number of people will be playing, which makes you more likely to win."

    4% picked this

  5. Sandy holds that the chances

    Bad Conclusion Match

    Sandy never talks about the chances of anyone winning. Her conclusion is about when it's optimal to play based on the size of the pot. She isn't ever claiming that your chances of winning change. She's implying that the potential payout of winning is bigger in some cases than others.

    17% picked this

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