Are you a good doctor? (Math Puzzle - A look at probability)

Here’s an interesting math puzzle involving probability…

1 out of every 10,000 citizens of a country has a deadly virus.  A blood test can be used to determine whether or not somebody has the virus.  The test is accurate 98% of the time.  There are 1 million people in the country, and each person is tested.  A man who tested positive for the virus talks to his doctor, who informs the man that there is a 98% chance that he does, in fact, have the virus.  The doctor is not correct in making this statement… what is the actual probability that the man has the deadly virus?

To see the answer, scroll down to the bottom.  If you would like help in understanding the logic behind the solution, keep reading.

First of all, we have to understand what probability is.

We know that 1 out of every 10,000 people actually have the virus.

Therefore, out of 1 million people, 100 people have the virus.  (1,000,000 ÷ 10,000 = 100)

The probability that a citizen of the country has the virus is 100 ÷ 1,000,000 = 0.0001 = 0.01%.

This probability is independent of the blood test.

Once the blood test is administered to every citizen in the country, the question becomes a bit more difficult.  We need to determine the probability that a person who tested positive for the virus does, in fact, have the virus.

So, how many people tested positive for the virus?

98% of people who have the virus tested positive.  98% of 100 is 98.

If 100 people have the virus, then 999,900 people do not have the virus. (1,000,000 – 100 = 999,900)

98% of people who don’t have the virus tested negative.  This means that 2% of people who don’t have the virus tested positive.  2% of 999,900 is 19,998.

The total number of people who tested positive is 98 + 19,998 = 20,096.

Now we can calculate the probability that the man who tested positive does, in fact, have the virus.

20,096 people tested positive for the virus.  Of these, 98 people actually have the virus.

Therefore, the probability that the man has the virus is 98 ÷ 20,096 = 0.0049 = 0.49%.

That’s less than half of one percent!

When considering this fact, it is tempting to think that the test is practically worthless – but this is not the case.

Before the test was administered, all we knew was that 0.01% (one 100^th of a percent) of the population had the virus.  Now a group has been isolated in which about half of a percent has the virus.

If this group is re-tested, the probability that a person testing positive does, in fact, have the virus will be about 24%.

If the group that tests positive a 2^nd time is re-tested, the probability that a person testing positive does, in fact, have the virus will be about 92%.

It’s amazing to see how quickly accuracy is gained through the process of re-testing.

And now you know a little bit of the mathematics that goes into being a good doctor.

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Watermelons and New Years' Parties (Puzzle)

Here’s a fun math puzzle involving percents.  The solution is not terribly difficult – but it is surprising how many people end up getting the answer wrong.  Why not give it a try:

The owner of a farm stand puts a bunch of watermelons on display at the beginning of the day.  The watermelons have a mass of 200 kg.  99% of the watermelons’ mass is water.  It is hot, and during the day water evaporates from the melons.  At the end of the day, 98% of the watermelons’ mass is water.  Assuming that no watermelons were sold during the day, what is the mass of the watermelons at the end of the day?

If you’d like to check your answer, the correct solution is below.

 

If you’re having a hard time, consider this puzzle.  The math involved the same as the watermelon problem, but the way in which the puzzle is presented tends to make the solution easier to find.

Some guys planned a New Years’ party and invited only women.  At the party, there were a total of 200 people, 99% of whom were women.  A lot of the women thought that the party was lame, so they left.  After one hour, 98% of the people left at the party were women.  How many total people were at the party after one hour?

This puzzle is slightly easier to grasp because it explicitly states that there are men and women.  (And, it states that only women leave the party.)

 

See below for the solutions.

 

 

 

Solution: New Years’ party puzzle

It’s not very hard to determine how many men and women there were at the beginning of the party:

Women:  99% of 200 = 198           Men:  1% of 200 = 2

Since only women left the party, there are still two men after one hour.  However, these two men now comprise 2% of the total party crowd.  2 is 2% of 100… therefore, there are 100 people at the party after one hour.

 

Solution: Watermelon puzzle

The watermelon puzzle is a little trickier simply because the problem doesn’t explicitly state that the watermelon is made up of water and solids.  Therefore, many people get hung up focusing on the water.

If 99% of the watermelon mass is water, then 1% is made up of solids.  1% of 200kg is 2 kg – so there are 2 kg of solids.  Solids do not evaporate, so at the end of the day, there are still 2 kg of solids.  This now comprises 2% of the total watermelon mass.  2 kg is 2% of 100 kg.  Therefore, there are 100 kg of watermelons left at the end of the day.

Happy New Years’, Everyone!

 

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Ready, Go, SET!

Riddles and puzzles are a good way to challenge yourself and increase your logic skills.  They also give valuable practice in solving the story problems that seem to haunt so many math students.

Here’s a brain teaser that can be solved with simple math.  (As is the case with many math puzzles, there is more than one way to solve the problem.)  If you think you’re up to it, try to answer the question before reading the solutions and explanations at the bottom of this page.

There are 340 players in a SET* tournament.  Each game in the tournament is played with 4 people.  For each game, there is one winner who moves on to the next round.  Rounds will continue until there is just one winner left – the champion.  In any round, if the number of players is not a multiple of 4, then some players will be chosen randomly to advance automatically so that the number of people playing in that round is divisible by 4.  (This will ensure that each game that is played will have 4 players.)  How many total games will be played in the tournament?

{Scroll down for solutions}

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*I love card games.  I also love activities that help people learn and/or increase their brain power.  SET is a card game that is simple to learn, fun for a wide range of players (recommended for players 6 years old and above), and helps build brain power.  As a teacher and card game enthusiast, I give SET my full recommendation.  (And now SET-Junior is available for younger players.)

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Solutions/Explanations

Method 1 (slow)

To solve the problem, many people get out pencil and paper and begin calculating the results, round by round.  For each round, they keep track of the number of games, and then add these up at the end to find the answer to the question.  This method can be quite time consuming.  One such solution is shown:

Method 2 (fast)

Of course, there is another way to solve this problem – a way that takes much less time.  In the previous solution, the focus was on the WINNERS from each round.  (The number of players in rounds 2 – 5 was determined by figuring out how many winners there were in the previous round and adding this amount to the number of people who were lucky enough to advance automatically.)  A much quicker solution involves focusing on LOSERS:

We know each game that is played will have 4 players.  This means that each game is guaranteed to have 3 losers.  The tournament begins with 340 players, and 339 of those players will be a loser at some point.  All we need to do is divide 339 by 3 to find our answer: There will be 113 games.

When looking at a tough problem, we often hurt ourselves by focusing on the wrong part of the problem.  Sometimes it is helpful to take a step back, reexamine the problem, and figure out if our focus is in the right place.


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