STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011
Review We have discussed counting techniques in Chapter 1. (Principle of counting, Permutation, Combination) Compute probabilities in certain situations.
Experiment A random experiment is a process whose outcome is uncertain. Example: Tossing a coin once or several times; Picking a card or cards from a deck; Measuring temperature of patients;
Events and Sample Spaces A sample spaces S of a random experiment is the set of all possible outcomes. An event E is any subset of the sample space S. Our objective is to determine P(E), the probability that event E will occur.
Example The experiment: Toss a coin 3 times. Sample space S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Examples of event include A={at least two heads}={hhh,hht,hth,thh} B={exactly two tails}={htt,tht,tth}
Example If two dice are rolled What is the sample space? What is the event?
Example If two dice are rolled What is the sample space? What is the event? S={(1,1),(1,2),...,(6,5),(6,6)} E={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
Sample Spaces with Equally Likely Outcomes: probability is easy to compute For many experiments, it is natural to assume that all outcomes in the sample space are equally likely to occur. If there are N(S) possible equally likely outcomes in S, then the probability assigned to each outcome is 1/N(S). If an event A consists of N(A) outcomes, then P(A) = N(A) N(S)
Example Recall the experiment where we toss a coin 3 times. Sample space S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Examples of event include A={at least two heads}={hhh,hht,hth,thh} B={exactly two tails}={htt,tht,tth} So what is the probability that there are at least 2 heads? What is the probability that there are exactly 2 tails?
Example If two dice are rolled, what is the probability that the sum of the upturned faces will equal 7?
Example A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women? What is the size of sample space? What is the size of the event?
Example A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women? What is the size of sample space? What is the size of the event? What is the probability of this event?
Example A deck of 52 playing cards is shuffled, and the cards are turned up one at a time until the first ace appears. Is the next card - that is, the card following the first ace - more likely to be the ace of spades or the two of clubs?
It is possible that the space has infinite number of outcomes: Toss a coins until you see a head, and record the coin toss history: The sample space of this experiment is S = {H, TH, TTH, TTTH, }. The sample spaces contains an infinite number of outcomes.
It is possible that the space has infinite number of outcomes: Toss a coins until you see a head, and record the coin toss history: The sample space of this experiment is S = {H, TH, TTH, TTTH, }. The sample spaces contains an infinite number of outcomes. And the outcomes are not equally likely to happen!
What do we do? Idea: Each outcome has some probability to happen For any event E, we add up the probability of the outcomes within that event, to compute the probability of E.
What do we do? Idea: Each outcome has some probability to happen For any event E, we add up the probability of the outcomes within that event, to compute the probability of E. Toss a coins until you see a head, and record the coin toss history: The sample space of this experiment is S = {H, TH, TTH, TTTH, }. What is the probability of H, TH, TTH respectively? What is the probability of the event: you finish the coin tosses with no more than 3 tosses?
To make the above idea formal, we need to define some concepts about sets.
Last Example: should not be hard A poker hand consists of 5 cards. If the card have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. For instance, a hand consisting of the five of spades, six of spades, seven of spades, eight of spades, and nine of hearts is a straight. What is the probability that one is dealt a straight?