A deck of cards is shuffled. You turn over the cards one by one.
1. What is the probability that at least one of each of the numbered denominations (1 - 10) is turned over before any of the twelve face cards is turned over?
2. What is the probability that at least one of each of the Jack, Queen and King is turned over before any numbered denomination (1 - 10) is turned over?
Before doing the calculation, which probability do you think is higher, and why? ... or are they the same?
For an extra challenge: Answer the same questions using a special 61-card deck consisting of 1 Ace, 2 Deuces, 3 Treys, through 10 tens, and 4 Jacks, 1 Queen and 1 King.
Extra Challenge #2: This is the only one that doesn't seem really really hard.
Call a face card F and a number card N. A card that can be anything call X.
There are four patterns that need to be considered:
FFFNA...
FFFFNA...
FFFFFNA...
FFFFFFN...
The N is to keep from overcounting. In each case we also need to ensure the several face cards at the front contains at least one of J,Q,K.
In the FFF case this probability is 4/20
In the FFFF case this probability is 6/15
In the FFFFF case the probability is 4/6
In the FFFFFF case we have all the face cards to it's guaranteed.
The four individual probabilities are then
6/61*5/60*4/59*55/58*4/20
6/61*5/60*4/59*3/58*55/57*6/15
6/61*5/60*4/59*3/58*2/57*55/56*4/6
6/61*5/60*4/59*3/58*2/57*1/56
Simplified separately
11/104371
22/1983049
55/83288058
1/55525372
Total simplifies to 673/5744004 or about 0.0001171656566
Edited on May 27, 2021, 7:41 am
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Posted by Jer
on 2021-05-26 11:21:18 |
Solution to extra challenge #2
