Let's imagine that we have a regular deck of cards in our hand. We shuffle it, and draw a random card from the deck. Let's model this with the means of probability calculation: let Ω = {ω = (l, m) : l ∈ {1, ..., 13}, m ∈ {heart, square, cross, spade}} and let P be a discrete uniform distribution with this set. (1) What is the probability that the card we draw is an ace (ie the number on the card is 1 – in this problem we don't care that sometimes the ace can also be 14)? In other words, calculate P(A) when A = {(1, m) : m ∈ {heart, diamond, club, spade}}. (2) Suppose we put the first card back in the deck, shuffle the deck again, draw a new card, and show it (without looking) to our friend. He tells us the card is a heart (and he's not lying). Given this information, what is the probability that the card is an ace? So, in other words, calculate P(A|B), where A is as above and B = {(l, heart) : l ∈ {1, 2, ..., 13}}. (3) Repeat the same trick as above. He tells us now (still not lying) that the number on the card is at most four. Given this information, what is the probability that the card is an ace? In other words, calculate P(A|C), where A is as above and C = {(l,m) : l ∈ {1,2,3,4},m ∈ {heart, square, club, spade} }.
Let's imagine that we have a regular deck of cards in our hand. We shuffle it, and draw a random card from the deck. Let's model this with the means of
(1) What is the probability that the card we draw is an ace (ie the number on the card is 1 – in this problem we don't care that sometimes the ace can also be 14)? In other words, calculate P(A) when A = {(1, m) : m ∈ {heart, diamond, club, spade}}.
(2) Suppose we put the first card back in the deck, shuffle the deck again, draw a new card, and show it (without looking) to our friend. He tells us the card is a heart (and he's not lying). Given this information, what is the probability that the card is an ace? So, in other words, calculate P(A|B), where A is as above and B = {(l, heart) : l ∈ {1, 2, ..., 13}}.
(3) Repeat the same trick as above. He tells us now (still not lying) that the number on the card is at most four. Given this information, what is the probability that the card is an ace? In other words, calculate P(A|C), where A is as above and C = {(l,m) : l ∈ {1,2,3,4},m ∈ {heart, square, club, spade} }.
Solution:
A deck has 52 cards. It contains 4 Ace cards.
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