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Three people all set down their identical notebooks on a table.
On the way out, they each randomly pick up one of the notebooks. What is the probability that none of the three people pick up the notebook that they started with?
That's correct probability!
What was the case?
Let's name peoples as Person - 1, Person - 2, Person - 3 and their notebooks as Notebook - 1, Notebook - 2 and Notebook 3 respectively.
Now there can be 6 ways 3 notebooks can be distributed among 3 persons like below.
(Here, for convenience, 3 different colors are assigned to the notebooks of 3 persons.)
As we can see, there are only 2 cases, where the each of person not getting own notebook. In rest of cases, at least 1 person got own notebook & notebooks of others are shuffled between 2.
So the probability that none of the three people pick up the notebook that they started with is 2/6 = 1/3.
Katherine and Zyan are playing a game using strange dice. Each die
is a cube with six sides. Katherine's die has sides numbered 3, 3, 3,
3, 3, and 6. Zyan's die has sides numbered 2, 2, 2, 5, 5, and 5.
To play the game, Katherine and Zyan roll their dice at the same time
and whoever rolls the higher value wins. If they play many times, who
will win more frequently, Katherine or Zyan?
This person will be winning more!
What was the game?
Shortest Way :
Katherine's die has sides numbered 3, 3, 3, 3, 3, and 6. And Zyan's die has sides numbered 2, 2, 2, 5, 5, and 5. That means whenever Zyan rolls a 2 then Katherine will always win. The probability that Zyan rolls 2 is 3/6 = 1/2. So in half of the cases, the Katherine will be winner of the game. Moreover, whenever, Zyan rolls a 5, Katherine can win if she rolls a 6. In short, in more than half cases, Katherine will win hence she has more advantage in this game.
Precise Way :
Katherine will win if
1. Zyan rolls a 2 with probability 3/6 = 1/2
2. Zyan rolls a 5 (probability 3/6 = 1/2) and Katherine rolls a 6 (probability 1/6). So the probability for this win = 1/12.
Hence, Katherine has got 1/2 + 1/12 = 7/12 chances of winning.
Zyan will win if he rolls a 5 (probability 1/2) and Katherine rolls a 3 (probability 5/6) with probability = 1/2 x 5/6 = 5/12.
That proves, Katherine has more chances (7/12 vs 5/12) of winning this game.
Another Way :
There can be 36 possible cases of numbers of cubes out of which in 15 cases Zyan seems to be winning and in 21 cases, Katherine is winning. Again, Katherine having more chances of winning (21/36 = 7/12) than Zyan (15/36 = 5/12).
