The Math behind Cracking Eggs!

The other day, a LuckyCat quiz called Don't Crack the Eggs! caught my attention. In this quiz you have to try not to click on the “fragile” eggs, that means that you have to choose all the eggs that are safe.

Methodology

The order of the eggs is random. The total cases are 30 while the favorable cases are 27 (3 eggs are bad).

Thus, the probability that the first egg is good is 27/30, or 90%

Another way to look at it is that if you play 30 games then you will probably lose on the first move a total of 3 times.

Probabilities

We have already seen that the probability that the first egg is safe is 90% Now that one egg is already safe, the total cases are 29 and the favorable cases are 26.

26/29

If we want to find out the probability that the first two eggs are safe we have to multiply the fractions.

27/30 x 26/29 = 80.68%

This means that the probability that the first two eggs are good is 80.68%

Now that our method is better to understand, let's fast forward a bit.

3 good eggs: 72.04%

4 good eggs: 64.03%

5 good eggs: 56.65%

6 good eggs: 49.85%

7 good eggs: 43.62%

8 good eggs: 37.93%

9 good eggs: 32.75%

10 good eggs: 28.07%

11 good eggs: 23.86%

12 good eggs: 20.09%

13 good eggs: 16.74%

14 good eggs: 13.79%

15 good eggs: 11.20%

16 good eggs: 8.96%

So, at this point, we already have picked 16 eggs and they were all good. That means we have 11 good eggs and 3 bad eggs left. Let's keep going!

17 good eggs: 7.04%

18 good eggs: 5.41%

19 good eggs: 4.06%

20 good eggs: 2.95%

21 good eggs: 2.06%

22 good eggs: 1.37%

23 good eggs: 0.86%

24 good eggs: 0.49%

25 good eggs: 0.24%

26 good eggs: 0.098%

and finally...

27 good eggs: 0.024%

To put it another way, to complete the quiz at 100% the approximate probability is 1 in 4,166.

Pretty cool, right?