Ever had a pistachio nut? Some of them are not cracked open and cannot be opened

Question

Ever had a pistachio nut? Some of them are not cracked open and cannot be opened by hand. In a quarter pound bag of pistachios, let X be the random variable representing the number of pistachios in the bag that cannot be opened by hand. The probability table is below: If two bags were selected one after the other, v/hat is the probability of finding that both bags had NO pistachios in the bag was too difficult to open by hand? If two bags were selected and the first bag was opened and had three pistachios in the bag that was too difficult to open. What is the probability that the second bag also had exactly three pistachios too difficult to open by hand? Suppose two bags of pistachios are selected at random. What is the probability that both bags have four or more pistachios too difficult to open by hand? Draw the bar graph of the probability distribution of the pistachio data at the left. Determine the mean of the distribution and label the location of the mean of the distribution. Determine the standard deviation of the distribution. Determine the location of the values that are one standard deviation on either side of the mean.

Explanation / Answer

Let X be the random variable representing the number of pistachios in th bag that cannot be opened by hand.

If two bags were selected at one after the other what is the probability of finding that both bags had no pistachios in the bag was too difficult to open by hand.

That is here we have to find P(bag1 has no pistachios in the bag was too difficult to open by hand. and bag2 has no pistachios in the bag was too difficult to open by hand.)

= P(bag1 no pistachios in the bag was too difficult to open by hand.) * P(bag2 has no pistachios in the bag was too difficult to open by hand.)

= P(X=0)*P(X=0 )= 0.5 * 0.5 = 0.25

If two bags were selected and the first bag was opened and had three pistachios in the bag that was too difficult to open what is the probability that the second bag has exactly three pistichios too difficult to open by hand.

P(X = 3) = 0.05

Suppose two bags are selected at random. What is the probability that both bags have four or more pistachios too difficult to open by hand?

P(bag1 have four or more pistachios too difficult to open by hand and bag2 have four or more pistachios too difficult to open by hand)

P(X >=4) * P(X >=4)

P(X = 4 or 5) * P(X = 4 or 5)

P(X=4)+P(X=5) * P(X=4)+P(X=5)

= 0.075+0.025 * 0.075+0.025

= 0.1*0.1 = 0.01

In the next part, we have to find mean and standard deviation the probability distribution.

mean = x*p(x)

variance = x2 *p(x) - mean2

sd = sqrt (variance)

The table for calculating mean and sd is, as follows.

mean = 1.03

variance = 2.9 - 1.032 =  1.839

sd = sqrt(1.839) = 1.36

x p xp x^2p 0 0.5 0 0 1 0.25 0.25 0.25 2 0.1 0.2 0.4 3 0.05 0.15 0.45 4 0.075 0.3 1.2 5 0.03 0.125 0.625 1.03 2.9

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