Binomial Probability Calculation in "Thinking, Fast and Slow"

Question

In the chapter "The law of small numbers" of "Thinking, Fast and Slow" there is the following example

Imagine a large urn filled with marbles. Half the marbles are red, half are white...From the same urn, two very patient marble counters take turns. Jack draws 4 marbles on each trial, Jill draws 7. They both record each time they observe a homogeneous sample— all white or all red. If they go on long enough, Jack will observe such extreme outcomes more often than Jill— by a factor of 8 (the expected percentages are 12.5% and 1.56%).

Kahneman, Daniel. Thinking, Fast and Slow (p. 110). Penguin Books Ltd. Kindle Edition.

So, I tried to check these number by using the Binomial Probability Calculator(https://stattrek.com/online-calculator/binomial.aspx)

And while 0.0625 / 0.0078125 = 8(exactly the number that the author mentioned)

The probabilities do not match..in fact the probabilities provided by the author are twice as much as ones provided by the calculator.

How to explain this difference? isn't the calculator correct?

Answer 1

You have to take into account that a homogeneous sample can be either full red or full white, contributing the factor of two.

Answer 2

The probability of $7$ reds is $(\frac{1}{2})^7$ and the same holds for $7$ whites so having an "extreme sample" has probability $$2 \cdot \frac{1}{2^7} =\frac{1}{2^6} = \frac{1}{64}= 0.015625$$ for 7 marbles.

For $4$ we get $$ 2 \cdot \frac{1}{2^4}= \frac{1}{8}=0.125$$

in accordance with the book. No binomial stuff needed, really.

The factor $8$ is just the $2^3$ from the quotient of $2^7$ and $2^4$.

Answer 3

$$(\frac{.0078125}{.0625})\times100=12.504%$$ Which means 4 marbles of same color are more likely to occur by 12.5 % compared to 7 marbles of same color. I don't know from where 1.56% came from.