Mario is testing out a new car at the racetrack. His grandmother took it for a mile lap and averaged 30mph. How fast does Mario have to drive in the second lap to average 60mph for the two miles?
90mph is an alluring answer, but it’s impossible. At 90mph the second lap would take 40 seconds. If Grannie’s first lap at 30mph took 2 minutes, traveling 2 miles in 2 minutes and 40 seconds is an average speed of 45mph. Traveling an average speed of 60mph for 2 miles would take 2 minutes and since the grandma’s lap took all of those 2 minutes, even if Mario could break the sound barrier he wouldn’t be able to average 60mph.
You have 12 black socks and 12 white socks mixed up in a drawer. You’re up very early and it’s too dark to tell them apart. What’s the smallest number of socks you need to take out (blindly) to be sure of having a matching pair?
Three socks. If the first sock is black, the second one could be black, in which case you have a matching pair. If the second sock is white, the third sock will be either black and match the first sock, or white and match the second sock.
You’re waiting to board your flight at the airport with 99 other passengers, each with an assigned seat. All but one of the passengers will gladly sit in their designated seat. The only exception is Randall, a scoundrel who refuses to follow the rules. When he boards, he will choose a random, unoccupied seat.
If a rule-following passenger finds someone in their spot, they will choose another one at a random from the remaining unoccupied seats.
What is the probability that the last person to board the plane will sit in their proper seat?
The randomness stops as soon as someone else sits in Randall’s assigned seat. The chances of this happening range from 1 out of 99 to 1 out of 1 (when only one seat remains).
Thus, the probability of the last person sitting in their own seat can be calculated as 1/99 plus the sum of 2 to 98 of the formula 1 / n × (n + 1), which works out to 0.5, or 50%.
So there’s a 50% chance the last passenger will sit in their own seat thanks to Randall for screwing up order and procedure when boarding an aircraft.
