Respuesta :
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
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
The PMF of the number of selecting basalt specimen is [tex]\boxed{\bf P=^{10}C_{x}(0.5)^{x}(0.5)^{10-x}}[/tex] and the probability of selecting all specimen of one of two type of rocks is [tex]\boxed{\bf 0.0325}[/tex].
Further explanation:
Concept used:
The probability of an event [tex]E[/tex] can be calculated as follpws:
[tex]\boxed{P(E)=\dfrac{n(E)}{n(S)}}[/tex]
Here, [tex]n(E)[/tex] is the number of favorable outcomes in an event [tex]E[/tex] and [tex]n(S)[/tex] is the number of element in sample space [tex]S[/tex].
The probability mass function of getting exactly [tex]r[/tex] success in [tex]n[/tex] independent trial in an experiment can be expresses as follows:
[tex]\boxed{P=^nC_{r}p^{r}q^{n-r}}[/tex]
Here, [tex]p[/tex] is the success probability and [tex]q[/tex] is the failure probability of an event.
Calculation:
A geologist has collected [tex]10[/tex] specimens of ballistic rock and geologist collected [tex]10[/tex] specimens of granite.
The total number of specimens is [tex]20[/tex].
The sample space is the collection of all possible outcomes in an experiment.
Therefore, the total possible outcomes are [tex]20[/tex].
The probability of selecting ballistic rock can be calculated as follows:
[tex]\begin{aligned}P_{b}&=\dfrac{10}{20}\\&=\dfrac{1}{2}\\&=0.5\end{aligned}[/tex]
The probability of selecting granite can be calculated as follows:
[tex]\begin{aligned}P_{g}&=\dfrac{10}{20}\\&=\dfrac{1}{2}\\&=0.5\end{aligned}[/tex]
The PMF of selecting exactly [tex]x[/tex] basalt specimen for analysis can be calculated as follows:
[tex]\boxed{P=^{10}C_{x}(0.5)^{x}(0.5)^{10-x}}[/tex]
Consider [tex]A[/tex] as an event that all rocks are selecting from one rock and [tex]n(A)[/tex] as the number of favorable outcomes in an event [tex]A[/tex].
The number of favorable outcomes in an event [tex]A[/tex] can be calculated as follows:
[tex]\begin{aligned}n(A)&=\left(^{10}C_{10}\cdot ^{10}C_{5}\right)+\left(^{10}C_{10}\cdot ^{10}C_{5}\right)\\&=\left(\dfrac{10!}{10!\cdot (10-10)!}\cdot \dfrac{10!}{5!\cdot (10-5)!}\right)+\left(\dfrac{10!}{10!\cdot (10-10)!}\cdot \dfrac{10!}{5!\cdot (10-5)!}\right)\\&=\dfrac{10\cdot 9\cdot 8\cdot 7\cdot 6}{5\cdot 4\cdot 3\cdot 2\cdot 1}+\dfrac{10\cdot 9\cdot 8\cdot 7\cdot 6}{5\cdot 4\cdot 3\cdot 2\cdot 1}\\&=252+252\\&=504\end{aligned}[/tex]
The total possible outcomes of selecting [tex]15[/tex] rocks in [tex]20[/tex] rocks can be calculated as follows:
[tex]\begin{aligned}n(S)&=^{20}C_{15}\\&=\dfrac{20!}{15!\cdot (20-15)!}\\&=\dfrac{20\cdot 19\cdot 18\cdot 17\cdot 16\cdot 15!}{15!\cdot 5!}\\&=\dfrac{20\cdot 19\cdot 18\cdot 17\cdot 16}{5\cdot 4\cdot 3\cdot 2\cdot 1}\\&=19\cdot 3\cdot 17\cdot 16\\&=15504\end{aligned}[/tex]
The probability [tex]P(A)[/tex] of selecting all specimen of one of two type of rocks can be calculated as follows:
[tex]\begin{aligned}P(A)&=\dfrac{504}{15504}\\&=0.0325\end{aligned}[/tex]
Thus, the probability that all specimens of one of the two types of rock are selected for analysis is [tex]\boxed{\bf 0.0325}[/tex].
Learn more:
1. Learn more about functions https://brainly.com/question/2142762
2. Learn more about numbers https://brainly.com/question/1852063
Answer details:
Grade: College school
Subject: Mathematics
Chapter: Probability
Keywords: Basalt specimen, geologist, rock, granite, PMF, probability mass function, exactly, probability, possible outcomes, favorable, failure success, , event, experiment, trial, P=n(E)/n(S).
