Probability Problems
Three coins are tossed at the same time. What is the probability that all three coins will come up heads OR all will come up tails?
1/4...
1/8 (all heads) + 1/8 (all tails) = 1/4
In the original Monty Hall Problem, which action would give you the highest chance of winning the prize? Should you switch, stick, or does it not matter what you choose to do?
Switch (2/3 chance of winning)
In what way is The Wall Game Show deterministic?
Once the ball is released, it only follows one path to the slot.
What kind of mathematics are involved in Monopoly besides just probability?
Markov Process & Linear Algebra (with the transition matrices)
In how many different ways can five students be seated in three chairs?
60... 5(4)(3)=60
The game show host shows you three doors, where one has the car prize and the other have two goats.
Here is what happens: You pick the door on the right. The host chooses to open the door in the middle AT RANDOM. As it turns out, that door was holding a goat. Now you have the option to switch. Should you switch, stick, or does it not matter what you choose to do?
Doesn't matter
* in the classic Monty Hall problem, the host will open a door with a probability of 1 (because Monty knows where the goat is)
* in this example, Monty does NOT know where the goat is so he will open the door with a goat with probability 1/2 (if you choose goat initially)
What kind of pattern is observed for the ball's movement down the Plinko board after the first ten episodes of the game?
Inverse Exponential Function
Which space is a player most likely going to be on in the long term?
Five students, all of different heights, are randomly arranged in a line. What is the probability that the tallest student will be first in line and the shortest student will be last in line?
1/20...
(1)(3)(2)(1)(1) = 6 ways in which students can be arranged under given conditions...
(5)(4)(3)(2)(1) = 120 ways under no conditions
Probability Calculation: 6/120 = 1/20
A host places 100 boxes in front of you, one of which contains $1000. You pick box 3. The host removes all the boxes except 3 and 42, stating that the money is in just one of the boxes. What is the probability that the money is in box 42?
A) 1% ... B) 50% ... C) 98% ... D) 99%
99% ... Here's the solution:
P($1000 in box 3) = 1/100
P($1000 is not in box 3) = 99/100..
When the 98 boxes are removed, the probability that it is in box 3 is still 1/100 and the probability that it is not in box 3 stays the same at 99/100.
How is the Wall an example of the Markov Process?
1) The mathematics involved takes into account a set of random variables
2) The future outcome depends only on the present state (the initial position of the ball) rather than previous outcomes.
Recall that Markov chains are mathematical systems that "hop" (transition) from one "state" (condition/state of variables) to another. To tally the transition probabilities, mathematicians use a "transition matrix." Think about it as a matrix of probabilities.
Which is the least frequently visited space?