A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. if the geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis, what is the pmf of the number of basalt specimens selected for analysis? what is the probability that all specimens of one of the two types of rock are selected for analysis?

Part 1

Given that there are 10 specimens of basaltic rock and 10 specimens of granute, the probability of selecting a basaltic rock is 10 / 20 = 0.5 and the probability of selecting a granite is 10 / 20 = 0.5

Thus, the probability mass function of the number of basalt specimens selected for analysis is given by

[tex]f(x)=\left(^{10}_x\right)(0.5)^x(0.5)^{10-x}[/tex]



Part 2

The probability that all specimens of one of the two types of rock are selected for analysis is given by the sum of the probabilities that 10 basalt specimens and 5 igneous specimen is selected and the probabilities that 5 basalt specimens and 10 igneous specimen is selected.


The probability that 10 basalt specimens and 5 igneous specimen is selected is given by

[tex]\frac{\left(^{10}_{10}\right)\left(^{10}_{5}\right)}{\left(^{20}_{15}\right)}=\frac{252}{15,504}=0.01625[/tex]

The probability that 5 basalt specimens and 10 igneous specimen is selected is also given by

[tex]\frac{\left(^{10}_{10}\right)\left(^{10}_{5}\right)}{\left(^{20}_{15}\right)}=\frac{252}{15,504}=0.01625[/tex]

Therefore, the probability that all specimens of one of the two types of rock are selected for analysis is given by 2(0.01625) = 0.0325