Help an English Major Solve a Math Problem?

Question

My brother-in-law, who is 18 and too smart for his own good, keeps telling everyone that the probability for any situation is 50/50. For example, if you ask him "will the Saints win the superbowl," his answer is, it's 50/50--which ignores the fact that they haven't gotten there. He says this is because either they will win or they will lose.

At nearly twice his age I am constantly frustrated with this line of reasoning, but am at a loss to explain why it is so wrong. Does it have to do with mutual exclusivity and probability? Help.

Answer 1

If the choice you've given him (in this case, the Saints) is included in the set of possible answers (in this case, the two teams competing for the Superbowl), then his "basic" probability of 50 percent is correct.

There are other factors that enter into the equation ... all the staticistics on home-vs-away matchups for both teams, record on real-vs-artifical turf, health and wellness of teams, offense-vs-defense matchups, and so on. At the end of the day, the oddsmakers will try to take all this into account and give folks a different percentage chance - maybe the Saints will be given 1-9 odds ... meaning they're about 90 percent likely to lose the game.

Of course, if the choice ISN'T part of the answer set, then the probability MUST BE ZERO (can't pick it if it can't occur, right?).

Alternatively, if the choice is the ONLY one in the answer set, the probability must be UNITY (it's the only choice there is!).

However, you can re-instruct him a little bit, using a simple six-sided die.

The chances of rolling a 1 are not 50 percent ... they're 1-in-6 (about 16.7 percent). The chance of rolling two 1s in a row is the mulitplication of the two probabilities (1-in-6 x 1-in-6 = 1-in-36, or about 2.8 percent).

A fair bit different than the oversimplified "it will, or it won't" probability theory he's proposing, isn't it?

Answer 2

For any given probability situation there is a set of outcomes and a probability associated with each outcome. Just because there are two possible outcomes does NOT mean that it is a 50/50 situation. Example:

Put two oranges and a pear into a paper bag and draw one fruit out at random, clearly there are 2 outcomes a pear or an orange. But the probability of getting an orange is 66 2/3% and
probability of getting a pear is 33 1/3%. Clearly not 50/50.

Answer 3

He confusing two outcomes with equal probability outcomes.

A coin toss necessarily has two possible outcomes. A 'fair' coin means that these two have the same probability. You can weight a coin to make it biased. As an extreme example, I've seen phony coins with two heads and not tails. In that case
P(head) = 1
P(tail) = 0
That's hardly 50/50. (Which mean two possibilities p=0.50 for each)


So, it's true that there's two possiblities:

1. they win the superbowl
or
2. they don't

There's little justification for saying that the probabilities are equal.

Sounds like he's ripe for you to take some of his money. There are several 'bets' that people who reason his way usually lose.

Answer 4

Just because there are two possibilities does not mean they have an equal likelihood of being correct. For example, the probability of a child being born a male or female is 50/50, but the probability of that child having a major physical deformity (like having three legs) is far less than it being normal and healthy. In that case, the probability would be based on the genetic structure of the parents, but could be estimated by the percent of the population. For the Super Bowl scenario, the probability is dependent on their statistics once they get there. If they don't get there, the probability that they will win is 0. I hope that helps.

Answer 5

No it isn't a 50/50 chance.

What he is trying to say is that there are 2 outcomes and that the Saints winning the superbowl is one of those outcomes. However this isn't so, even if the chance of winning everygame is 50/50 and they have 3 games to win then the chances will be

0.50x0.50x0.50 (i.e. 50% chance in each game)
so there is a 12.5% chance in winning.

However this is extremely over-simplified. The chances in each game would be extremely hard to calculate. And yes is does involve mutual exclusivity and probability. Unfortunatly it would take an essay to explain calculating the probability and also the explanation of why it isn't a 50/50 chance. Just hope this helps a little.

(Btw. I live in the UK so I have no idea about how superbowl works or how many games you play)

Answer 6

He's not so smart. Let's say we have a coin. I flip it 100 times. By some odd chance, the first 99 times my coin ends up heads. Now, what is the chance that it will be heads once more? Well, it's 50/50 (sure, there is the minute possibility it lands on its edge). Football games are not coins, however. Your brother-in-law is confusing possible outcomes with the possibility of the outcome. Yes, the Saints will either win or lose: 50/50. But that has nothing to do with projecting the odds that they will either win or lose. This is something completely different.

Answer 7

He is probably confused about what 50/50 means. It's not that there are two outcomes, but that the outcomes are equally likely to occur. For instance, I might win the lottery jackpot or I might not. There are two outcomes, but certain factors make one outcome much more likely than the other.

If there is no confusion about what 50/50 means and he insists that the probability of a certain team winning is the same as the probability of it losing, try this thought experiment with him. Suppose 10 teams can win the Super Bowl. What is the probability that a certain team does win it? While each team has two outcomes (win or lose), it is impossible for them all to have 50/50 chances of winning.

Answer 8

He is absolutely wrong.

Probability has to do with populations, not individual outcomes.

The easiest way to look at it is this: take a bag of 99 pennies and a pound coin. Any coin taken will be either a penny or a pound, but the pennies come from a population of 99 and the pound from a population of 1.

In this case a single choice is 99 times more likely to be a penny than a pound. If he doesn;t agree, offer him odds on it and then take your chances - you will win 98% of the time.

Answer 9

To say the Saints will win the Superbowl among 12 teams to choose currently, 32 teams to choose from the beginning of season, I would say it wouldn't be 50/50. It would only makes sense to say 50/50 in this situation would be assuming there is ONLY one team other than the Saints who is equally talented to play against them in the Superbowl. Same analogy to it's either win or lose the lottery, so the chance is 50/50. It's only 50/50 if you bought half of all possible combinations of numbers. I hope I'm explaining it clear enough.

Answer 10

if the super bowl was decided by a coin toss it would be 50/50 since the coin has no recollection of its past flips- if you flip a coin 99 times and it lands on heads every time the probability of it landing on tails the next time is still 50/50. he is trying to compare a "random walk" outcome with a dynamic situation (ie:play by play of each team) that has many more variables such as first line players, weather, location, etc. since each team has probably studied past plays of the other team and adapts strategy during the game..this is the equivalent of the "efficient market" where flaws are recognized and taken advantage of leaving few opportunities on the table. i guaruntee if a sports bookie offered 50/50 odds on every game they would quickly go bankrupt..