False Negatives and Cancer.

In the voiceover, Cologuard claims it finds 92% of colon cancers. The ad also provides a generously large and easy legible banner stating that false positives and false negative results can occur. 

The next banner gives more detail.

If your grandparent gets a "Yes" response, what is the probability that they do have colon cancer? If they get a "No", what are the chances they don't have it?

https://www.ispot.tv/ad/op0r/cologuard-finding-things

If you've never paid much attention to advertisements before, this is pretty amazing. Usually the fine print is illegible even on a 62" ultra hi-definition tv paused at just the right moment.

So ... thank you Cologuard for being honest and up-front.

514: Probability Game

Temporary return! A quick probability question ...

Player A's score is determined by taking the highest of 3 dice.
Player B's is determined by taking second-highest of 8.
Who wins more games?

Player A's score is determined by taking the highest of 3 dice.
Player B's is determined by taking second-highest of 8.
Who wins more games?

— Ben Orlin (@benorlin) June 16, 2016

512: Factory Ratios 3

We started with this:

In a factory, the ratio of men to women is 2:3. The ratio of right-handed men to left-handed men is 7:3. The ratio of right-handed women to left-handed women is 11:1. What fraction of the factory workforce is right-handed?
And then we extended with: What is the fewest number of employees possible in this building?

If the ugly-sweater party was held in that factory with the minimum number of workers possible and the  ratio of red to green ugly sweaters was 7:13, how likely is it that a left-handed man wore a red ugly sweater?

Source:

New GCSE Maths question: Ratio and proportion is the buzz thing! How are your students going to tackle it? pic.twitter.com/RyPsmc9z4m

— m4ths.com (@m4thsdotcom) December 16, 2015

507: Orderly Probability

Scenario 1: Which winning number group is more likely to occur?

1-2-3-4-5-6  OR 4-8-15-16-23-42

Scenario 2: Which winning number group is more likely to occur if the numbers are drawn in any order and THEN put into ascending order by the presenter?

1-2-3-4-5-6  OR 4-8-15-16-23-42

In which of the above two scenarios is getting the winning numbers more likely?

source: Jeff Suzuki

502: Powerful Question

It's not included in the PEMDAS Order of Operations ...

Should $a^{b^c} = ({a^b})^c$ or should it be $a^{b^c} = a^{(b^c)}$ ??

Does $3^{2^0}$ equal 1 or 3?
 
Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?

For the record,  $a^{b^c} = a^{(b^c)}$ is the accepted order of operations here.

499: Blue-Eyed Blondes

If the proportion of blonds among blue-eyed people is greater than among the population as a whole, is it also true that the proportion of blue-eyed people among blonds is greater than among the population as a whole?

source.

465: Probability Machine

You've all seen Plinko and its variations.  Here's one:

This one has a normal curve drawn on the background. Should it be a normal curve or more triangular like the arrangement of the pins?

Would it still be in this shape if the pins were arranged in a rectangle?

433: Do We Need to Feed the Meter?

If you are parked during any part of the time indicated, then you need to pay, even if it's only for a few minutes at the beginning or end of your shopping. (If you only are there for the five minutes between 8:00 and 8:05, you only pay a pro-rated fee.)


If you park at any random time and leave your car there for an hour, are you more likely to pay or more likely to not pay?


source.

431: Mr Smith has Two Children.

Thank you for reminding me of this:

Mr Smith has two children. One of them is a boy. What is the probability the other child is also a boy?
— solve my maths (@solvemymaths) April 4, 2015

407: The More Likely Sequence

Skimming the MTBoS Finds Interesting Things video from the Global Math department.

Which will be more likely to occur first

in a string of coin tosses, HTH or HTT?

This is the kind of question that becomes difficult because of assumptions made by the listener, assumptions that change the problem subtly and thus change the resulting expected probability. The listener/reader here naturally changes this question to "Which group of three is more likely?" That's not what he asked.
 
We should refer back to the Monty Hall question.
Source: Bob Lochel, Obvious Debate. from TEDTalk by Peter Donelly.

355: Millionaire

Would you guess? $250,000 if correct and $100,000 if she refuses to try for it.

349: Hole-in-One Insurance

If the average golfer is able to get a hole-in-one once in approximately 3000 rounds of golf (18 holes apiece), then what is the probability of any one of 100 average golfers getting a hole-in-one on the 5th hole during the weekend golf tournament?

What's the best way to find this out if you're the insurance company that will write this policy?

345: Fair or Foul?

Sullivan bought a die at the magic shop. He rolls it 155 times and gets the following results:

• ONE: twenty-eight times
• TWO: twenty times
• THREE: fifteen times
• FOUR: thirty-one times
• FIVE: thirty-two times
• SIX: twenty-nine times.

Is this die loaded?
What is the probability he will get a 6 on the next roll?

344: Monty Hall

Once upon a time, the world's smartest person (Marilyn vos Savant, IQ: 228) received a question for her newspaper column …

 Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? Craig. F. Whitaker, Columbia, MD

Marilyn's answer was surprising to many people. What do you think?

307: Expected Value vs Psychology

Which would you press ... and why?

269: Probability Sums

5 distinct numbers are chosen at random from {1,2,3,4,5,6,7,8,9}.

p(k) = probability their sum = k.

What are some of the ways you can find this in general?
What sum is/are the least likely?
Which sum is/are most likely?

p(15)=?

p(35)=?

source:

5 distinct numbers are chosen at random from {1,2,3,4,5,6,7,8,9}. p(k) = probability their sum = k. p(15)=?,..., p(35)=?
— Republic of Math (@republicofmath)

and

Five distinct nmbrs chosen at random from {1,2,3,...,9} and added together. What is the most likely sum?
— James Tanton (@jamestanton)

183: Italian Roulette

Could this work? What might go wrong with this plan?

If you divided the pizza into EIGHT pieces, what would be your chances of having to pay?

167: Probability, p5

Here's some more advice for you. Is there anything true about this advice?

Snakes on a plane:
When you're flying, always take a pet snake with you in your hand luggage. The probability of there being TWO snakes on the plane is almost zero, so you will be safe from snake attack.

Staying dry at the cricket match:
Follow the example of the famous mathematician Hardy and take an umbrella with you to cricket matches. If you forget your umbrella it is more likely to rain, so if you remember to take it with you it is more likely to be sunny all day.

 

166: Lottery, p4

Here's some more advice for gamblers. Do you agree with it or not?


Coin Flipping:
If tails has come up on the last 9 occasions then it's a good idea to call tails again.

Winning at Roulette:
If red has come up lots of times in a row, you should bet on black next.

165: Lottery, p3

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.As we all know, there's lots of free advice on how to win the lottery, usually given out by someone who, strangely, seems more willing to sell the information to you instead of using it themselves.  Odd, that. Anyway, here's some advice you just bought for a small fee:

Roughly equal numbers of odd and even are drawn most weeks, so you should pick a good mixture of odds and evens.

Never choose six numbers all from the same group - for example, all single digits, all multiples of five, all with the same last digit ...

Always pick some higher numbers from the 30s and 40s. 

 

What do you think? Advice worth paying for?