WOO logo

Wizard Recommends

Ask The Wizard #420

On average, how many people do you need to randomly select to get three people with a matching birthday?

Ace2

The answer is approximately 88.7389 days.

 

Here is my solution (PDF).

This question is asked and discussed in my forum at Wizard of Vegas.

George Wendt (the actor who played Norm on Cheers) died on May 20, the same day of the year as the last episode of Cheers aired 32 years before. What are the odds?

anonymous

Let me rephrase the question as what the odds he would die on the anniversary of the show's finale? The answer is simple 1 in 365.25. If you pick two days randomly with replacement from the calendar the probability they will be the same is the inverse of the number of days in a year, or 1/365.25.

Is there a way to estimate the number of primes under any given number?

anonymous

Yes! According to the Prime Number Theorem, the number of primes under n can be estimated as n/ln(n).

According to the theorem, the average distance between primes from 1 to n can be estimated as ln(n). Thus, the number of primes under n can be estimated as n/ln(n).

An extension of this theorem is the nth prime number can be estimated as n*ln(n).

For example, the number of primes under one billion can be estimated as 1,000,000,000/ln(1,000,000,000) =~ 48,254,942. The actual number of primes under one billion is 50,847,534. So, the estimate is 5.1% too low. This degree of error decreases as n increases.

It takes five days to paddle a canoe down a river with the current. It takes seven days to paddle a canoe up the same river against the current.

  1. How long would it take with no current?
  2. How long would it take to get down the river with the usual current without a paddle?

anonymous

1. 35/6 days

2. 35 days

 

Recall that distance = rate × time.

Let define some variables:

  • d = distance
  • c = rate of canoe
  • f = rate of the flow of the river

Let's put into equations the two pieces of information we're given:

  • d = (c+f)*5
  • d = (c-f)*7

Let's equate the two equations:

(c+f)*5 = (c-f)*7

5c + 5f = 7c - 7f
12f = 2c

c = 6f

Let's arbitrarily choose f=1. Then c=6*1=6. Also d=(6+1)*5 = 35.

To answer the first question let's solve for t when c=0: 35=6t.

t=35/6 = 5 5/6 days.

To answer the second question let's solve for t when c=0 and f=1: 35=1t.

t=35/1 = 35 days.

This question is also asked and solve in the Mind Your Decisions YouTube channel in the video titled Most US college students cannot solve this.