Mathematics, 31.03.2020 01:58 brittanyb70

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?

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Answer from: catycait27p5rc5p

(a) 0.8649

(b) 0.6469

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

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.

Answer from: Quest

step-by-step explanation:

Answer from: Quest

yes

step-by-step explanation:

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