Chapter 15: Probability
- What is Probability?
The probability of an event or events occurring is the long run chance of the event occurring.
It is often expressed as a percentage, proportion or fraction, and can be thought of as the percentage of times an event would occur if many trials were performed.
If outcomes areequally likely, then the probability of an event can be calculated as:
P(Event) = Count of all possible ways an event can occur
Count of all possible outcomes
Example 1. If I have a standard deck of 52 cards, what is the probability I pick out an ace?
Answer: P(Ace) = __# of ways I can get an ace____ = _4_
# of all possible cards I could get 52
2. Independent or not independent? (The and rule)
- The probability that two events A and B occursimultaneously is:
P(A and B) = P(B|A) x P(A) = P(A|B) x P(B)
- P(A|B) denotes the conditional probability of A given we know the event B has occurred.
- If A and B are independent, then the probability that A occurs does not depend on whether B occurs (and vice versa). Then P(A|B)=P(A) and P(B|A)=P(B),
and the equation above becomes
P(A and B) = P(B) x P(A) = P(A) x P(B)
Example 2.
- If I pick out two cards from a deck without looking at either, what is the probability that the first card is a black ace and the second card is another black card?
- If I pick out two cards, look at the first and see that it’s a black ace, what is the probability that the second card is a black card?
- If I pick out a card, look at it, then return it to the deck and pick another card, what is the probability that I get a black card both times?
Answers:
- P(1st card black ace AND 2nd card a black card)
= P(1st card a black ace) x P(2nd card a black card | 1st card a black ace)
= _2_ x 25 = _50_
52 51 2652
- P(2nd is black | first was black ace) = 25/51
- P(1st card black AND 2nd card black) = P(1st card black)xP(2nd card black)
= 26_ x 26 = _1_
52 52 4
3. Disjoint or not disjoint? (The OR rule)
- The general addition rule is:
P(A or B) = P(A) + P(B) - P(A and B)
- Two events are disjoint if they cannot happen simultaneously.
- If two events are disjoint, then we do not need to include the - P(A and B) in the rule above, since P(A and B) equals 0.
- Note that two events being disjoint implies that the two events are dependent, since knowing that A happens tells you something about B: it tells you that B did not happen. But if two events are dependent, they may or may not be disjoint.
a)Since picking a card that is both black and red is impossible, the two ‘events’ are disjoint. Since picking a black card means that the card could not be red, the two events are dependent.
b)If I told you that I picked a card higher than a 7, then the probability that this card is a 10 depends on this information. However, getting a ten means that both of these events took place (they are not disjoint).
Example 3. If I pull one card from the deck what is the probability that …
- I get an ace or a king?
- I get an ace or a black card?
Answers:
- P(ace or king) = P(ace) + P(king)
= _4_ + 4_ = _2_
52 52 13
- P(ace or black) = P(ace) + P(black) – P(ace and black)
= _4_ + 26 - _2_
52 52 52
= _28_
52
4. Complements
- The complement of an event is the probability that it did not occur:
P(Ac) = 1 – P(A)
Example 4. Say I pick three cards from a deck (without replacement). What is the probability that at least one of the three cards is larger than 2 (assuming aces are higher than kings)?
Answer: P(at least one of the three cards is larger than 2) = 1-P(three cards all equal 2)
= 1 – P(1st card = 2) x P(2nd card = 2 | 1st card = 2) x P(3rd card =2 | 1st and 2nd cards = 2)
= 1 - _4_ x 3 x 2_
52 51 50
= 5524
5525
Practice Questions
When you’re faced with a question, you must first think: is it ….
a)conditional?
b)OR; if so, are they disjoint?
c)AND; if so, are they independent?
d)Would it be easier to use the complement rule? (Are there too many combinations to count if you don’t use the complement rule…i.e. maybe you should use it?)
1)If I pull one card from a standard deck of 52 cards, what’s the probability I get a jack, queen, or king of hearts given that it is a red card?
2)If you are dealt three cards from a deck, one at a time, then:
a)What is the probability that at least one is a jack?
b)What is the probability that they are all jacks?
c)What is the probability that the third card is an ace given the first two were not aces?
d)You get all black cards?
e)The first spade you get is on the third card?
3)Repeat question 2, except treat the cards as replaced! If you are dealt three cards from a deck, one at a time, then:
a)What is the probability that at least one is a jack?
b)What is the probability that they are all jacks?
c)What is the probability that the third card is an ace given the first two were not aces?
d)You get all black cards?
e)The first spade you get is on the third card?