Chapter 14 Week 5, Monday

Introductory Example Consider a fair coin: Question: If I flip this coin, what is the probability of observing heads? Answer: Everyone knows the answer is 50%, but let’s look closer at what this actually means.

Introductory Example Consider a fair coin: Trial 1: I flip a coin and observe a head. trial observationH So Far: 100% of my trials produced heads.

Introductory Example Consider a fair coin: Trial 2: I flip a coin and observe a tail. trial observationHT So Far: 50% of my trials produced heads.

Introductory Example Consider a fair coin: Trial 3: I flip a coin and observe a tail. trial observationHTT So Far: 33% of my trials produced heads. (this is called the “relative frequency”)

Introductory Example Consider a fair coin: Probability: Has to do with long-term behavior.

Introductory Example Consider a fair coin: Law of Large Numbers: For “independent trials”, as the number of trials increases, the long-run relative frequency gets really close to a single value (in this case 50%)

Some Vocabulary Heads Tails Trial 1 Heads Tails Trial 2 Heads Tails Trial 3 Trial: Each occasion upon which we observe a random phenomenon Outcome: The value of the random phenomenon Sample Space: The collection of all possible outcomes

More Complicated Example Consider two fair coins: Question: If I flip these coins, what is the probability of observing 1 head and 1 tail? Answer: Not as obvious as before. The true likelihood is 50%.

More Complicated Example Consider two fair coins: Each Trial Heads, Heads Tails, Tails Heads, Tails Tails, Heads If either of these two outcomes occur, then we observed 1 heads and 1 tails

More Complicated Example Consider two fair coins: Each Trial Heads, Heads Tails, Tails Heads, Tails Tails, Heads We call a group of outcomes an “Event”. What is the probability of this event?

More Complicated Example Consider two fair coins:

A Nice Assumption If you assume that every outcome in the sample space has the same probability of occurring, then you can calculate the probability of an event occurring through a formula! Probability of Event A = (Number of outcomes in A) / (Total number of outcomes) Each Trial Heads, Heads Tails, Tails Heads, Tails Tails, Heads P[1 heads and 1 tails] = 2/4 = 50% P[at least 1 heads] = 3/4 = 75% P[no tails] = 1/4 = 25%

Another Example Consider a fair die: Each Trial P[at least 4] =3/6 = 50% P[more than 4] =2/6 = 33% P[5] =1/6 = 16.5% P[either 2, 3, or 6] =3/6 = 50% P[more than 2 AND less than 4] =1/6 = 16.5%

Probability Properties (1)For any event, “A”, P[A] is between 0% and 100% (2)Consider the event, “S”, consisting of all possible outcomes. P[S]=1 (3)For any event, “A”, consider the event “not A” (denoted: A C ). Then: P[A C ]=100%-P[A] (4)For any events, “A” and “B”: P[A or B] = P[A] + P[B] – P[A and B]

Probability Properties Each Trial Consider a fair die: P[1, 2, 3, 4, 5, or 6] =6/6 = 100% P[1 or 2] = 2/6 = 33% P[not {1 or 2}]= P[{1 or 2} C ] = 1 – P[1 or 2] = 100% – 33% = 67% P[{1,2,5,6} or {1,2,3}] = P[1,2,5,6]+P[1,2,3] - P[{1,2,5,6} and {1,2,3}] = (4/6) + (3/6) – (2/6) = 5/