Answers: Distributions and probability

Answers to odd numbered problems

Homework 3: Distributions & probability

1. Theoretical probability — we can apply a known theory to determine probability of an event. For example, Mendel’s theory allows us to predict expected probability given certain types of crosses. Mendel’s famous F2 3:1 ratio applies to one gene, two alleles with complete dominance. For systems that apply, out of 100 F2 offspring, 75 expected to be dominant phenotype and 25 expected to have recessive phenotype.
Empirical probability — no applicable theory available, probability of an event estimated from history. For example, lifetime risk of lung cancer by smoking is about 1 in 15 or 0.07.

3. Types
rate: i, ii, iv
proportion: iii
ratio: v, vi, vii

5. Sample with replacement means you draw a “card” then replace the card to the deck before drawing another “card.” It means each time you draw a card all cards have an equal chance to be selected and a card may be selected more than once.

7. Here’s my results. I used
sample(5, 1, replace = TRUE)

I assigned 1 to O, 2 to +, and so on to 5 = star.
Then, for my guesses (actual) trials

1. O (2)      6. star (5) == match!
2. O (4)      7. star (2)
3. O (2)      8. + (5)
4. star (2)   9. waves (2)
5. star (4)  10. waves (5)

esp
0/10   1     6/10 210
1/10  10     7/10 120
2/10  45     8/10  45
3/10 120     9/10  10
4/10 210    10/10   1
5/10 252

sum(1,10,45,120,210,252,210,120,45,10,1) = 1024 possible answers

9. Question expect use of normal table. Rcmdr: Distributions → Continuous → Normal → Normal probabilities
1.5 times sd, the value is 8 (= mean + 1.5*sd)

pnorm(c(8), mean=5, sd=2, lower.tail=FALSE) = 0.0668072

11. between -0.75 and +0.25 times greater than the standard deviation? Mean µ = 5 and σ = 2
calculate area greater than -0.75 and subtract area less than +0.25 = between 3.5 and 5.5 = 0.7733726-0.5987063 = 0.1746663

13. 50 percent. That’s the consequence of “normal distribution:” exactly half will be less than the mean, half will be greater than the mean.