1) 2 possible outcomes
2) Fixed number of trials
3) Independent events
4) Probability is constant
The Binomial Formula
n = number of trials
x = number of successes
"At most" means "less than or equal to". Ex: If I wanted at most 5, I am interested in the probabilities between 0 and 5. [p(0)+p(1)+p(2)+p(3)+p(4)+p(5)]
"At least" means "greater than or equal to". Ex: If I wanted at least 5 in a sample of ten, I am interested in the probabilities between 5 and 10. [p(5)+p(6)+p(7)+p(8)+p(9)+p(10)]
6.2 # 13-18, the experiment is to toss a fair coin three times. Use the binomial formula to find the indicated probabilities.
13. No heads were observed.
p = 0.50 , n = 3, x = 0.
p(0) = 3C0 (0.5)0 (1-.0.5)(3-0)
p(0) = (1) (1) (0.125)
p(0) = 0.125
14. One head was observed.
p = 0.50 , n = 3, x = 1.
p(1) = 3C1 (0.5)1 (1-.0.5)(3-1)
p(1) = (3) (0.5) (0.25)
p(1) = 0.375
15. Two heads were observed
p = 0.50 , n = 3, x = 2.
p(2) = 3C2 (0.5)2 (1-.0.5)(3-2)
p(2) = (3) (0.25) (0.5)
p(2) = 0.375
16. Three heads were observed.
p = 0.50 , n = 3, x = 3.
p(3) = 3C3 (0.5)3 (1-.0.5)(3-3)
p(3) = (1) (0.125) (1)
p(3) = 0.125
17. At most two heads were observed.
"At most" refers to a combined probability, in this case it's "less than or equal to" two heads. So we are interested in the cumulative probability from 0 to 2, which we already found in the previous problems so...
p(0) + p(1) + p(2) = ?
0.125 + 0.375 + 0.375 = 0.875
18. More than two heads were observed
"More than" refers to a combined probability, in this case it's "greater than" two heads. So we are interested in the probabilities greater than 3, however for this problem set there is only one such probability that matches this...
p(3) = 0.125
22. Probability that at least 3 of the next 4 dentists surveyed will recommend sugarless gum. Assume the recommendation is given 95% of the time.
"At least" refers to a combined probability, in this case we're interested in a minimum of 3 successes. So we need to find probability of observing a 3 and the probability of observing a 4.
p(3) = 4C3 (0.95)3 (1-0.95)(4-3)
p(3) = (4) (0.857375) (.05)
p(3) = 0.1715
p(4) = 4C4 (0.95)4 (1-0.95)(4-4)
p(4) = (1) (0.81450625) (1)
p(4) = 0.8145
p(3+4) = 0.8145 + 0.1715
p(3+4) = 0.8145 + 0.1715
p(3+4) = 0.9860
If you are unclear on any of this, please watch this video.
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