Question

A diamond can be classified as either gem dash quality or industrial dash grade. Suppose that 93​% of diamonds are classified as industrial dash grade. ​(a) Two diamonds are chosen at random. What is the probability that both diamonds are industrial dash grade​? ​(b) Six diamonds are chosen at random. What is the probability that all six diamonds are industrial dash grade​? ​(c) What is the probability that at least one of six randomly selected diamonds is gem dash quality​? Would it be unusual that at least one of six randomly selected diamonds is gem dash quality​?

Answer 1

Answer:

(a) 0.8649

(b) 0.6469

(c) 0.353

Step-by-step explanation:

We are given that a diamond can be classified as either gem dash quality or industrial dash grade. Suppose that 93​% of diamonds are classified as industrial dash grade.

(a) Two diamonds are chosen at random.

The above situation can be represented through Binomial distribution;

$P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....$

where, n = number of trials (samples) taken = 2 diamonds

            r = number of success = both 2

           p = probability of success which in our question is % of diamonds

                  that are classified as industrial dash grade, i.e; 0.93

LET X = Number of diamonds that are industrial dash grade​

So, it means X ~ $Binom(n=2, p=0.93)$

Now, Probability that both diamonds are industrial dash grade is given by = P(X = 2)

       P(X = 2)  = $\binom{2}{2}\times 0.93^{2} \times (1-0.93)^{2-2}$

                      = $1 \times 0.93^{2} \times 1$

                      = 0.8649

(b) Six diamonds are chosen at random.

The above situation can be represented through Binomial distribution;

$P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....$

where, n = number of trials (samples) taken = 6 diamonds

            r = number of success = all 6

           p = probability of success which in our question is % of diamonds

                  that are classified as industrial dash grade, i.e; 0.93

LET X = Number of diamonds that are industrial dash grade​

So, it means X ~ $Binom(n=6, p=0.93)$

Now, Probability that all six diamonds are industrial dash grade is given by = P(X = 6)

       P(X = 6)  = $\binom{6}{6}\times 0.93^{6} \times (1-0.93)^{6-6}$

                      = $1 \times 0.93^{6} \times 1$

                      = 0.6469

(c) Here, also 6 diamonds are chosen at random.

The above situation can be represented through Binomial distribution;

$P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....$

where, n = number of trials (samples) taken = 6 diamonds

            r = number of success = at least one

           p = probability of success which is now the % of diamonds

                  that are classified as gem dash quality, i.e; p = (1 - 0.93) = 0.07

LET X = Number of diamonds that are of gem dash quality

So, it means X ~ $Binom(n=6, p=0.07)$

Now, Probability that at least one of six randomly selected diamonds is gem dash quality is given by = P(X $\geq$ 1)

       P(X $\geq$ 1)  = 1 - P(X = 0)

                      =  $1 - \binom{6}{0}\times 0.07^{0} \times (1-0.07)^{6-0}$

                      = $1 - [1 \times 1 \times 0.93^{6}]$

                      = 1 - $0.93^{6}$ = 0.353

Here, the probability that at least one of six randomly selected diamonds is gem dash quality​ is 0.353 or 35.3%.

For any event to be unusual it's probability is very less such that of less than 5%. Since here the probability is 35.3% which is way higher than 5%.

So, it is not unusual that at least one of six randomly selected diamonds is gem dash quality​.