Mathematics : asked on bri9263
 12.03.2020

A fair coin is flipped three times. Let X be the number of heads observed. (a) Give both the possible values and probability mass function of X. (b) Find P(X ≥ 1) and P(X > 1). (c) Compute E[X] and V ar(X).

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24.06.2023, solved by verified expert
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a)

The possible values are 0, 1, 2 and 3.

The probability mass function of X is:

P(X = 0) = 0.125

P(X = 1) = 0.375

P(X = 2) = 0.375

P(X = 3) = 0.125

b)

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20, A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

c) E(X) = 1.5, Var(X) = 0.75

Step-by-step explanation:

For it time the coin is flipped, there are only two possible outcomes. Either it comes up heads, or it comes up tails. Each toss is independent of other tosses. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

In which A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20 is the number of different combinations of x objects from a set of n elements, given by the following formula.

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

And p is the probability of X happening.

The expected value of the binomial distribution is:

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

The variance of the binomial distribution is:

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times.

This means that A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

Fair coin means that it is equally as likely to be heads or tails, so A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

a) Give both the possible values and probability mass function of X.

The possible values are from 0 to n, so 0, 1, 2 and 3.

The probability mass function is the probability of each outcome. So

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

(b) Find P(X ≥ 1) and P(X > 1).

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

(c) Compute E[X] and Var(X).

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

A fair coin is flipped three times. Let X be, №17887571, 12.03.2020 03:20

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Mathematics
Step-by-step answer
P Answered by PhD

a)

The possible values are 0, 1, 2 and 3.

The probability mass function of X is:

P(X = 0) = 0.125

P(X = 1) = 0.375

P(X = 2) = 0.375

P(X = 3) = 0.125

b)

P(X \geq 1) = 0.875, P(X  1) = 0.5

c) E(X) = 1.5, Var(X) = 0.75

Step-by-step explanation:

For it time the coin is flipped, there are only two possible outcomes. Either it comes up heads, or it comes up tails. Each toss is independent of other tosses. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

The expected value of the binomial distribution is:

E(X) = np

The variance of the binomial distribution is:

V(X) = np(1-p)

A fair coin is flipped three times.

This means that n = 3

Fair coin means that it is equally as likely to be heads or tails, so p = \frac{1}{2} = 0.5

a) Give both the possible values and probability mass function of X.

The possible values are from 0 to n, so 0, 1, 2 and 3.

The probability mass function is the probability of each outcome. So

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 0) = C_{3,0}.(0.5)^{0}.(0.5)^{3} = 0.125

P(X = 1) = C_{3,1}.(0.5)^{1}.(0.5)^{2} = 0.375

P(X = 2) = C_{3,2}.(0.5)^{2}.(0.5)^{1} = 0.375

P(X = 3) = C_{3,0}.(0.5)^{3}.(0.5)^{0} = 0.125

(b) Find P(X ≥ 1) and P(X > 1).

P(X \geq 1) = P(X = 1) + P(X = 2) + P(X = 3) = 0.375 + 0.375 + 0.125 = 0.875

P(X  1) = P(X = 2) + P(X = 3) = 0.375 + 0.125 = 0.5

(c) Compute E[X] and Var(X).

E(X) = np = 3*0.5 = 1.5

Var(X) = np(1-p) = 3*0.5*0.5 = 0.75

Mathematics
Step-by-step answer
P Answered by PhD
Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram
Mathematics
Step-by-step answer
P Answered by PhD

Cost of 7 gallons=$24.50

Cost of 1 gallon=24.50/7=3.5

Cost of 15 gallons=15*3.5=52.5

Cost of 15 gallons will be $52.5

Mathematics
Step-by-step answer
P Answered by PhD

The answer is in the image 

The answer is in the image 
Mathematics
Step-by-step answer
P Answered by PhD

y=2x+15

where y=Value of coin

x=Age in years

Value of coin after 19 years=2*19+15

=$53

Therefore, Value after 19 years=$53

Mathematics
Step-by-step answer
P Answered by PhD

F=ma

where F=force

m=mass

a=acceleration

Here,

F=4300

a=3.3m/s2

m=F/a

    =4300/3.3

    =1303.03kg

Mathematics
Step-by-step answer
P Answered by PhD

Salesperson will make 6% of 1800

=(6/100)*1800

=108

Salesperson will make $108 in $1800 sales

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