Answer:
a) 0.371
b) 0.561
Step-by-step explanation:
We can answer both questions using conditional probability.
(a) We need to calculate the probability of obtaining two aces given that you obtained at least one. Let's call <em>A</em> the random variable that determines how many Aces you have. A is a discrete variable that can take any integer value from 0 to 4. We need to calculate
Since having 2 or more aces implies having at least one, the event is equal to the event
. Therefore, we can rewrite the previous expression as follows
We can calculate each of the probabilities by substracting from one the probability of its complementary event, which are easier to compute
We have now to calculate P(A = 0) and P(A = 1).
For the event A = 0, we have to pick 13 cards and obtain no ace at all. Since there are 4 aces on the deck, we need to pick 13 cards from a specific group of 48. The total of favourable cases is equivalent to the ammount of subsets of 13 elements of a set of 48, in other words it is . The total of cases is
. We obtain
For the event A = 1, we pick an Ace first, then we pick 12 cards that are no aces. Since we can pick from 4 aces, that would multiply the favourable cases by 4, so we conclude
Hence,
We conclude that the probability of having two aces provided we have one is 0.371
b) For this problem, since we are guaranteed to obtain the ace of spades, we can concentrate on the other 12 cards instead. Those 12 cards have to contain at least one ace (other that the ace of spades).
We can interpret this problem as if we would have removed the ace of spades from the deck and we are dealt 12 cards instead of 13. We need at least one of the 3 remaining aces. We will use the random variable B defined by the amount of aces we have other that the ace of spades. We have to calculate the probability of B being greater or equal than 1. In order to calculate that we can compute the probability of the <em>complementary set</em> and substract that number from 1.
In order to calculate P(B=0), we consider the number of favourable cases in which we dont have aces. That number is equal to the amount of subsets of 12 elements from a set with 48 (the deck without aces). Then, the amount of favourable cases is . Without the ace of spades, we have 51 cards on the deck, therefore
We can conclude
The probability to obtain at least 2 aces if we have the ace of spades is 0.561