Random Puzzles 

Workings

Hint
Look at the second letter in each word.

Answer
Julie.

The second letters of the names are the vowels in alphabetical order.

Workings

Hint

Answer
23.

Reasoning
By likely, we mean greater than 50% chance.

With one person there is a 0 percent chance that you'll have two people with the same birthday.

With two people the probability that they won't share a birthday is 364 / 365. The probability that they will share a birthday is therefore 1 - (364 / 365).

With three people the probability that they won't share a birthday is the same as for two people, times 363 / 365. So the probability that three people will share a birthday is 1 - (364 / 365) x (363 / 365).

Notice that with each additional person added, the probability that they shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining.

We keep adding people until the %age is greater than 50%.

When we have 23 people the %age is 50.729%.

Workings

Hint

Answer
6423: 109191 / 17. The numbers 1234567890 have been replaced by 7182435069 respectively.

Workings

Hint

Answer
1680.

This is n to the power of 5, starting with 2^5 = 32, with the last digit missing, 7^5 = 16807.