## Probably more likely than probable – Reblog

This is a reblog from here Probably more likely than probable // Revolutions

What kind of probability are people talking about when they say something is “highly likely” or has “almost no chance”? The chart below, created by Reddit user zonination, visualizes the responses of 46 other Reddit users to “What probability would you assign to the phase: <phrase>” for various statements of probability. Each set of responses has been converted to a kernel destiny estimate and presented as a joyplot using R.

Somewhat surprisingly, the results from the Redditors hew quite closely to a similar study of 23 NATO intelligence officers in 2007. In that study, the officers — who were accustomed to reading intelligence reports with assertions of likelihood — were giving a similar task with the same descriptions of probability. The results, here presented as a dotplot, are quite similar.

For details on the analysis of the Redditors, including the data and R code behind the joyplot chart, check out the Github repository linked below.

Github (zonination): Perceptions of Probability and Numbers

## n sweets in a bag, some are orange…

The other day in the news there was a note about a particular question in one of the national curriculum exams… I thought it was a bit of an odd thing for a maths question to feature in the news and so I thought of having a look a the question. Here it is:

There are $n$ sweets in a bag.

6 of the sweets are orange.

The rest of the sweets are yellow.

Hannah takes at random a sweet form the bag. She eats the sweet.

Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hanna eats two orange sweets is $\frac{1}{3}.$

a) Show that $n^2-n-90=0$

It sounds like an odd question, but after giving it a bit of thought it is actually quite straightforward; and I am glad they ask something that makes you think, rather than something that is purely a mechanical calculation.

So, let’s take a look: Hannah is taking sweets from the bag at random and without replacement (she eats the sweets after all). So we are told that there are 6 orange sweets, so at the beginning of the sweet-eating binge, the probability of picking an orange sweet is:

$\displaystyle P(\text{1 orange sweet})=\frac{6}{n}$.

Hannah eats the sweet, remember… so in the second go at the sweets, the probability of an orange sweet is now:

$\displaystyle P(\text{2nd orange sweet})=\frac{5}{n-1}$.

Now, they tell us that the probability of eating two orange sweets is $\frac{1}{3}$, so we have that:

$\displaystyle \left( \frac{6}{n} \right)\left( \frac{5}{n-1} \right)=\frac{1}{3}$,

$\displaystyle \frac{30}{n^2-n}=\frac{1}{3}$,

$\displaystyle n^2-n=90$,

which is the expression we were looking for. Furthermore, you can then solve this quadratic equation to find that the total number of sweets in the bag is 10.

The only thing we don’t know is if the sweets are just orange in colour, or also in flavour! We will have to ask Hannah!