Respuesta :
Answer:
a) 0.789, this is the complement of the probability of no repeat offenders; b) 0.411; c) 0.033; d) μ = 1.429; e) σ = 9.58
Step-by-step explanation:
For part a,
The probability that no parolees are repeat offenders is 0.211. This means the probability of at least one is a repeat offender is the complement of this event. To find this probability, subtract from 1:
1-0.211 = 0.789.
For part b,
To find the probability that 2 or more are repeat offenders, add together the probability that 2, 3, 4 or 5 parolees are repeat offenders:
0.216+0.162+0.032+0.001 = 0.411.
For part c,
To find the probability that 4 or more are repeat offenders, add together the probabilities that 4 or 5 parolees are repeat offenders:
0.032+0.001 = 0.033.
For part d,
To find the mean, we multiply each number of parolees by their probability and add them together:
0(0.211)+1(0.378)+2(0.216)+3(0.162)+4(0.032)+5(0.001)
= 0 + 0.378 + 0.432 + 0.486 + 0.128 + 0.005 = 1.429
For part e,
To find the mean, we first subtract each number of parolees and the mean to find the amount of deviation. We then square it and multiply it by its probability. Then we add these values together and find the square root.
First the differences between each value and the mean:
0-1.429 = -1.429;
1-1.429 = -0.429;
2-1.429 = 0.571;
3-1.429 = 1.571;
4-1.429 = 2.571;
5-1.429 = 3.571
Next the differences squared:
(-1.429)^2 = 2.0420
(-0.429)^2 = 0.1840
(0.571)^2 = 0.3260
(1.571)^2 = 2.4680
(2.571)^2 = 6.6100
(3.571)^2 = 12.7520
Next the squares multiplied by the probabilities:
0(2.0420) = 0
1(0.1840) = 0.1840
2(0.3260) = 0.652
3(2.4680) = 7.404
4(6.6100) = 26.44
5(12.7520) = 63.76
Next the sum of these products:
0+0.1840+0.652+0.7404+26.44+63.76 = 91.7764
Lastly the square root:
√(91.7764) = 9.58
Probabilities are used to determine the outcomes of events.
- The probability that one or more are repeat offenders is 0.789
- The probability that two or more are repeat offenders is 0.411
- The probability that four or more are repeat offenders is 0.033
- The standard deviation of repeat offenders is 1.093
- The expected number of repeat offenders is 1.429
The table is given as:
[tex]\left[\begin{array}{ccccccc}x &0 &1 &2 &3 &4 &5 &P(x) &0.211 &0.378 &0.216& 0.162 &0.032 &0.001\end{array}\right][/tex]
(a) Probability that one or more are repeat offenders
This is represented as: [tex]P(x \ge 1)[/tex]
Using the complement rule, we have:
[tex]P(x \ge 1) = 1 - P(x = 0)[/tex]
So, we have:
[tex]P(x \ge 1) = 1 - 0.211[/tex]
[tex]P(x \ge 1) = 0.789[/tex]
The probability that one or more are repeat offenders is 0.789
(b) Probability that two or more are repeat offenders
This is represented as: [tex]P(x \ge 2)[/tex]
Using the complement rule, we have:
[tex]P(x \ge 2) = 1 - P(x = 0) - P(x = 1)[/tex]
So, we have:
[tex]P(x \ge 2) = 1 - 0.211 - 0.378[/tex]
[tex]P(x \ge 2) = 0.411[/tex]
The probability that two or more are repeat offenders is 0.411
(c) Probability that four or more are repeat offenders
This is represented as: [tex]P(x \ge 4)[/tex]
So, we have:
[tex]P(x \ge 4) = P(x = 4) + P(x = 5)[/tex]
[tex]P(x \ge 4) = 0.032 + 0.001[/tex]
[tex]P(x \ge 4) = 0.033[/tex]
The probability that four or more are repeat offenders is 0.033
(d) The expected number of repeat offenders
This is calculated as:
[tex]\mu = \sum x \times P(x)[/tex]
So, we have:
[tex]\mu = 0 \times 0.211+ 1\times 0.378 + 2 \times 0.216 + 3 \times 0.162 + 4 \times 0.032 + 5 \times 0.001[/tex]
[tex]\mu = 1.429[/tex]
The expected number of repeat offenders is 1.429
(e) The standard deviation
This is calculated as:
[tex]\sigma= \sqrt{\sum (x^2 \times P(x)) - \mu^2}[/tex]
[tex]\sum (x^2 \times P(x))[/tex] is calculated as:
[tex]\sum (x^2 \times P(x)) = 0^2 \times 0.211+ 1^2 \times 0.378 + 2^2 \times 0.216 + 3^2 \times 0.162 + 4^2 \times 0.032 + 5^2 \times 0.001[/tex]
[tex]\sum (x^2 \times P(x)) = 3.237[/tex]
So, we have:
[tex]\sigma= \sqrt{\sum (x^2 \times P(x)) - \mu^2}[/tex]
[tex]\sigma = \sqrt{3.237 - 1.429^2}[/tex]
[tex]\sigma = \sqrt{1.194959}[/tex]
[tex]\sigma = 1.093[/tex]
The standard deviation of repeat offenders is 1.093
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