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tatyana61 [14]
3 years ago
6

A burglar alarm system has six fail-safe components. the probability of each failing is 0.05. find the probability that exactly

three will fail. round to the nearest thousandth.
Mathematics
1 answer:
insens350 [35]3 years ago
4 0
<span>0.002 For 6 systems with exactly 3 failures, you need to calculate 6 pick 3 or 6!/(3!3!) = 720/(6*6)=720/36=20 different combinations of 3 systems to fail. Now the probability of each of those failures will be the probability of the subsystems not failing times the probability of the subsystems fail. So 0.95^3 * 0.05^3 for 3 subsystem not failing time 3 subsystems failing. So 0.95^3 * 0.05^3 = 0.857375*0.000125 = 0.000107172. And multiply by the 6 pick 3 calculated earlier, gives. 0.000107172 * 20 = 0.002143438 And rounding to the nearest thousandth, gives 0.002</span>
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Ice cream usually comes in 1.5-quart boxes (48 fluid ounces), and ice cream scoops hold about 2 ounces. However, there is some v
Troyanec [42]

Answer:

(a) The expected value and standard deviation of the amount of ice cream served at the party are 54 ounces and 1.25 ounces respectively.

(b) The expected value and standard deviation of the amount of ice cream left in the box after scooping out one scoop are 46 ounces and 1.031 ounces respectively.

(c) Because the variance of each variable is dependent on the other.

Step-by-step explanation:

The random variable <em>X</em> and <em>Y</em> are defined as follows:

<em>X</em> = amount of ice cream in the box

<em>Y</em> = amount of ice cream scooped out

The information provided is:

E (X) = 48

SD (X) = 1

V (X) = 1

E (Y) = 2

SD (Y) = 0.25

V (Y) = 0.0625

(a)

The total amount of ice-cream served at the party can be expressed as:

<em>X</em> + 3<em>Y</em>.

Compute the expected value of (<em>X</em> + 3<em>Y</em>) as follows:

E(X+3Y)=E(X)+3E(Y)\\= 48+(3\times2)\\=48+6\\=54

Compute the variance of (<em>X</em> + 3<em>Y</em>) as follows:

V(X+3Y) = V (X)+3^{2}V(Y)+2\times 3Cov (X,Y)\\=1+(9\times0.0625)+0\\=1.5625

Then the standard deviation of (<em>X</em> + 3<em>Y</em>) is:

SD(X + 3Y) =\sqrt{V(X + 3Y)}\\\sqrt{1.5625}\\=1.25

Thus, the expected value and standard deviation of the amount of ice cream served at the party are 54 ounces and 1.25 ounces respectively.

(b)

The amount of ice-cream left in the box after scooping out one scoop is represented as follows:

<em>X</em> - <em>Y</em>.

Compute the expected value of (<em>X</em> - <em>Y</em>) as follows:

E(X-Y)=E(X)-E(Y)\\=48-2\\=46

Compute the variance of (<em>X</em> - <em>Y</em>) as follows:

V(X - Y) =V(X)+V(Y)-2Cov(X,Y)\\=1+0.0625-0\\=1.0625

Then the standard deviation of (<em>X</em> - <em>Y</em>) is:

SD(X-Y) =\sqrt{V(X -Y)}\\\sqrt{1.0625}\\=1.031

Thus, the expected value and standard deviation of the amount of ice cream left in the box after scooping out one scoop are 46 ounces and 1.031 ounces respectively.

(c)

The variance of the sum or difference of two variables is computed by adding the individual variances. This is because the variance of each variable is dependent on the others.

7 0
3 years ago
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