MATH 243, LECTURE 11
1. Using probability rules
Let’s recall the basic rules for discrete probability problems.
•If there are Doutcomes in the sample space which are a priori equally likely, then the chance of
achieving one of Nof these outcomes is N
D.
•P(Adoesn’t happen) = 1 −P(A).
•If event Aand event Bhave no outcomes in common,
P(Aor B) = P(A) + P(B).
•If the outcome of event Xis unrelated to the outcome of event Y(they are independent) then
P(X=Aand Y=B) = P(X=A)×P(Y=B)
Example 1. Find the probability of getting exactly one four when rolling a die three times.
Example 2. Find the probability that you there is a pair dealt in a hand with five cards.
2. Continuous (vs. discrete) random variables
.
Let’s remember how to find probabilities of flipping some number of heads in a sequence of coin flips.
As the number of flips gets large, the natural of the questions change.
For example, with 80 flips, we probably won’t get exactly 40 heads. The probability of that is .089 But
P(35 ≤Y≤45) = .781 and P(30 ≤Y≤50) = .982, so the probability of getting “near 30 heads” is high.
As the number of trials gets very large, the probability of any one outcome gets very small. It starts
being less relevant to ask for example what is the probability that you would get exactly 31 heads out of
80. The answer is just too small to matter. What works better is asking the chances that the number
of heads will be in some range. This change of emphasis (from “individual” measurement to ranges) is
important when we look at continuous variables.
•If a random variable Xtakes on a finite number of possible values, it is called a discete random
variable. All of our examples so far have been discrete.
•If a random variable Xtakes on a range of possible values, it is called a continuous random
variable.
•The sample space is still the collection of possible values for X.
•The probability distribution of a random variable is the assignment of probabilities to the values
in the sample space.
•It no longer makes sense to ask what P(X=a) is, but rather what is P(a≤X≤b).
Example 3. Let Xbe a number randomly chosen between 0and 1. (Note that it is hard to randomly
choose numbers in this fashion.)
•What is P(X=.3)
•If we assume all choices between 0and 1are equally likely, what is P(X≥.5).
•What is P(X≤.3or X≥.6)?
Example 4. What is the probability that a number chosen between 0 and 3 lies between 1.5 and 2.4?
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