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Six hundred paving stones were examined for cracks, and 15 were found to be cracked. The same 600 stones were then examined for discoloration, and 27 were found to be discolored. A total of 562 stones were neither cracked nor discolored. One of the 600 stones is selected at random. a. Find the probability that it is cracked, discolored, or both b. Find the probability that it is both cracked and discolored c. Find the probability that it is cracked but not discolored P(A Ո BC)

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Probability that a stone is cracked = 15/600; denote this event by [tex]A[/tex].

Probability that a stone is discolored = 27/600; denote this event by [tex]B[/tex].

Probability that a stone is neither cracked nor discolored = 562/600; in terms of [tex]A[/tex] and [tex]B[/tex], this event is captured by [tex](A\cup B)^c[/tex].

a. We want to find [tex]P(A\cup B)[/tex]. Use the known probability of its complement:

[tex]P(A\cup B)=1-P(A\cup B)^c=\dfrac{38}{600}[/tex]

b. We want to find [tex]P(A\cap B)[/tex]. By the inclusion-exclusion principle,

[tex]P(A\cap B)=P(A)+P(B)-P(A\cup B)=\dfrac{15+27-38}{600}=\dfrac4{600}[/tex]

c. By the law of total probability,

[tex]P(A)=P(A\cap(B\cup B^c))=P(A\cap B)+P(A\cap B^c)[/tex]

[tex]\implies P(A\cap B^c)=\dfrac{15-4}{600}=\dfrac{11}{600}[/tex]

Probability is the ratio of the favorable event to the total number of events.

Thus,  the values of part

a. 0.025, 0.045, 0.025

b. 0.025

c. 0.068178

Given;

15 stones were crack

27 were discolored

562 stones were neither crack nor discolor

And the total no. of stones is 600.

Find the probability that it is cracked, discolored or both

To find the probability we have

Probability

[tex]\rm Probability = \dfrac{favarable\ event}{total\ no.\ of\ event} [/tex]

a.  Probability that it is cracked, discolored, or both

1.  Favorable condition for one is cracked = 15

then

[tex]\rm P(crack) = \dfrac{total\ no.\ of\ crack\ item}{total\ no.\ of\ stones} \\\\ P(crack) = \dfrac{15}{600} \\\\ P(crack) = 0.025[/tex]

2. Favorable condition for discolored = 27

then

[tex]\rm P(discolored) = \dfrac{total\ no.\ of\ discolored\ stone}{total\ no.\ of\ stones} \\\\ P(discolored)= \dfrac{27}{600}\\\\ P(discolored)=0.045[/tex]

3. Favorable condition for both crack and discolored = 15

[tex]\rm P(cracked\ and\ discolored)=\dfrac{total\ no.\ of\ crack\ and\ discolored}{total\ no.\ of\ stones}\\\\ P(cracked,discolored)=\dfrac{15}{600} \\\\ P(cracked,discolored)=0.025[/tex]

b.  Favorable condition for both crack and discolored = 15

[tex]\rm P(cracked,discolored)=\dfrac{total\ no.\ of\ crack\ and\ discolored}{total\ no.\ of\ stones}\\\\ P(cracked,discolored)=\dfrac{15}{600} \\\\ P(cracked,discolored)=0.025[/tex]

c.  Favorable condition for cracked but not discolored

then total no. stone will be 573

[tex]\rm P(cracked,not\ discolored)=\dfrac{total\ no.\ of\ crack\ but\ not\ discolored}{total\ no.\ of\ stones}\\\\ P(cracked,discolored)=\dfrac{15}{573} \\\\ P(cracked,discolored)=0.06178\\ [/tex]

To learn more about probability link is given below.

https://brainly.com/question/795909

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