HSS 2381 Study Guide - Midterm Guide: Binomial Theorem, Binomial Distribution, To Tell The Truth
15 Dec 2016
School
Department
Course
Professor
HSS2381 - Quantitative Methods in Health Sciences I
Class 3a - Probability and the binomial distribution
1. A high-level view of probability
2. The binomial theorem
3. Generating a probability distribution
4. Area under the curve
1. A high-level view of probability:
- Probability of occurrence: calculated as likelihood of an event; short term is probability
- # of ways an event can occur divided by the total # of possible outcomes.
1. Probability: notation and limits:
- The probability of an event is expressed as = p(event) & ranges from 0 – 1.0
1. Probability: examples
a) ½ b) 1/6 c) ½ d) ¼ e) 4/52=1/13
1. Probability of any event:
- 2 alternatives when you toss a coin; heads or tails
- Sum of the probabilities of all individual events always equals 1.0
- p(heads) = 0.5 & p(tails) = 0.5, so p(heads)+p(tails) = 1.0
- In other terms, p(any outcome) is another way of saying some outcome must occur
2. The binomial theorem:
- The binomial theorem helps quantify the likelihood of events.
- Possible outcomes if coin tossed 2x? Where heads=H and tails=T:
HH
HT
TH
TT
- There are 4 possible outcomes considering order of outcomes, only 3 outcomes if only considering
head/tail
- 1 way to get 2 heads
- 1 way to get 2 tails; &
- 2 ways to get 1 head & 1 tail
2. The binomial theorem: three tosses
- If you tossed the coin 3x, there are 8 possible ways an outcome can occur:
find more resources at oneclass.com
find more resources at oneclass.com
1. HHH
2. HHT
3. HTH
4. THH
5. TTH
6. THT
7. HTT
8. TTT
2. The binomial theorem: four tosses = possible outcomes are
Outcome Specific outcomes Ways
4 heads (4H) HHHH 1
3H1 THHHT, HHTH, HTHH, THHH 4
2H2 THHTT, HTHT, HTTH,THHT, THTH, TTHH 6
1H3 TTTTH, TTHT, THTT,HTTT 4
4 tails (4T) TTTT 1
Total 16
2. The binomial theorem: detecting a pattern
- A pattern, # of outcomes is always 1 more than # of tosses & total # of ways the outcomes can occur is
the number of alternatives (heads/tails) raised to the power of tosses
# of Tosses # of Outcomes Total # of ways the outcomes can occur
1 2 2
2 3 4
3 4 8
4 5 16
3. Generating a probability distribution:
- Single coin toss has 2 possible outcomes (head/tail) w/ unbiased coin, probability on any trial of
obtaining head & tail (not-head) is 0.5
3. Possible outcomes of multiple coin tosses:
- Tossed unbiased coin 10x; possible outcomes shown & consider how many ways each outcome occur
3. Example: outcome of multiple coin tosses
- Only 1 possible way to get 10 heads if you toss unbiased coin 10 times (n=10)
- Probability of obtaining head in 1 coin toss is p=0.5
- The probability of obtaining 10 heads in 10 coin tosses is calculated as:
or
- 2= 1024; cuz only 1 way to obtain 10 head in 10 tosses, probability of obtaining 10 heads in 10 tosses
= 1/1024 or p=.00098
- In table, there are 10 ways where outcome can be 9 heads and 1 tail.
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Class 3a - probability and the binomial distribution: a high-level view of probability: Probability of occurrence: calculated as likelihood of an event; short term is probability. # of ways an event can occur divided by the total # of possible outcomes: probability: notation and limits: The probability of an event is expressed as = p(event) & ranges from 0 1. 0: probability: examples, b) 1/6 c) d) e) 4/52=1/13, probability of any event: 2 alternatives when you toss a coin; heads or tails. Sum of the probabilities of all individual events always equals 1. 0. P(heads) = 0. 5 & p(tails) = 0. 5, so p(heads)+p(tails) = 1. 0. In other terms, p(any outcome) is another way of saying some outcome must occur: the binomial theorem: The binomial theorem helps quantify the likelihood of events. There are 4 possible outcomes considering order of outcomes, only 3 outcomes if only considering head/tail. 1 way to get 2 tails; &
