Statistics 200 Lab

1

Lab 3: Review on ProbabilityPre-reading

  1. 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.

  1. 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?
  2. 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?
  3. 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:

  1. 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

  1. P(2nd is black | first was black ace) = 25/51
  1. 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 …

  1. I get an ace or a king?
  2. I get an ace or a black card?

Answers:

  1. P(ace or king) = P(ace) + P(king)

= _4_ + 4_ = _2_

52 52 13

  1. 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