Adding and Multiplying Probabilities - Handout | STAT 2, Study notes of Statistics

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STAT 2

Lecture 16:

Adding and multiplying

probabilities

Review: probability ● Frequentist theory of probability: if you repeat the event millions of time, what percentage will turn out a certain way ● Subjective theories of probability also exist ● In all theories: P(event happens) + P(event doesn't happen) = 100%

Example: the deck of cards ● After dealing the first two cards, there are 3 queens left out of 50 cards ● Probability is 3/50 = 6% ● This is a conditional probability

Example: the deck of cards ● I deal three cards in succession from a shuffled deck. What's the probability that the first card is the queen of spades, the second is the six of diamonds, and the third is a queen?

Example: the deck of cards ● P( st card is queen of spades, 2 nd is six of diamonds, 3 rd is queen) = 1/52 * 1/51 * 3/ = 3/132600 or 0.0023% ● Multiplication rule: to find the probabilities of successive events, multiply conditional probabilities given the previous events

Today ● Independent events ● Mutually exclusive events and the addition rule ● Trees and Venn diagrams

Independent events: definition ● Two events are independent if the outcome of one event doesn't change the probabilities of the outcomes of the other event

Independent events: example I toss a coin twice. ● First event: the first toss is heads or tails. Second event: the second toss is heads or tails Whether the first toss is heads or tails doesn't change the probability of heads or tails on the second toss, so the events are independent

Independent events: example ● If I toss a coin twice: P(two heads) = P( st toss head and 2 nd toss head) = P( st toss head) * P( nd toss head | 1 st toss head) by multiplication rule = P( st toss head) * P( nd toss head) = ½ * ½ = ¼

Dependent events: example I deal 2 cards from a shuffled deck. ● First event: the first card is a king ● Second event: the second card is a queen How do we show these events are not independent?

The “and” formula ● To find the joint probability of two events A and B: P(A and B) = P(A) * P(B|A) = P(B) * P(A|B) ● If A and B are independent: P(A and B) = P(A) * P(B)

II

Mutually exclusive events

The addition rule ● If two (or more) events are mutually exclusive, we can find the probability that one or the other of them happens by adding the probabilities of each of them

Example: the addition rule ● What is the probability the first card is a king or a queen? ● “ 1 st card king” and “ st card queen” are mutually exclusive ● P(king or queen) = P(king) + P(queen) = 4/52 + 4/52 = 8/52 = 2/