Question

(1) A company that produces fine crystal knows from experience that 10% of its goblets have cosmetic flaws and must be classified as "seconds." (a) If 6 goblets are randomly selected, what is the probability that only one is considered a second? (b) If 6 goblets are randomly selected, what is the probability that at least 2 are seconds? (c) Find EX, VX, and SDX a lab assistant to randomly select 8 of the specimens for analysis. (2) A geologist collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs (a) What is the probability that the random sample will contain exactly 5 granite rocks? (b) What is the probability that all specimens of one of the two types of rock are selected for analysis?

Answer 1

1) Here the random variable X is defined as the number of goblets which are classified as "second" in a random sample of 6 goblets.

So, X ~ Binomial (6, 0.1)

(a) The required probability is P[X = 1] = = 0.354294.

(b) The required probability is P[X >= 2] = 1 - P[X = 0] - P[X = 1] = 1 - $\binom{6}{0}0.1^00.9^6$ - $\binom{6}{1}0.1^10.9^5$

= 1 - 0.531441 -- 0.354294

=  0.822853

(c) E[X] = n*p = 6*0.1 = 0.6

V[X] = n*p*(1-p) = 6*0.1*0.9 = 0.54

SD[X] = $\sqrt{V[X]}$ = 0.7348469

2) Here the random variable Y is defined as the number of granite rocks in a random sample of 8 rock.

So, Y ~ Binomial(8, 0.5)

(a) The required probability is P[Y = 5] = $\binom{8}{5}0.5^50.5^3$ = 0.21875

(b) Here the required probability is

P[all specimens of one of the two types of rock are selected]

= 1 - P[either all are basaltic or all are granite]

= 1 - ( P[all are basaltic] + P[all are granite] )

= 1 - P[all are basaltic] - P[all are granite]

= 1 - $\binom{8}{0}0.5^00.5^8$ - $\binom{8}{8}0.5^80.5^0$

= 1 - 0.00390625 - 0.00390625

= 0.9921875.