1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
dangina [55]
3 years ago
13

Suppose n people, n ≥ 3, play "odd person out" to decide who will buy the next round of refreshments. The n people each flip a f

air coin simultaneously. If all the coins but one come up the same, the person whose coin comes up different buys the refreshments. Otherwise, the people flip the coins again and continue until just one coin comes up different from all the others. a) What is the probability that the odd person out is decided in just one coin flip? b) What is the probability that the odd person out is decided with the kth flip? c) What is the expected number of flips needed to decide odd person out with n people?
Mathematics
1 answer:
blondinia [14]3 years ago
7 0

Answer:

Assume that all the coins involved here are fair coins.

a) Probability of finding the "odd" person in one round: \displaystyle n \cdot \left(\frac{1}{2}\right)^{n - 1}.

b) Probability of finding the "odd" person in the kth round: \displaystyle n \cdot \left(\frac{1}{2}\right)^{n - 1} \cdot \left( 1 - n \cdot \left(\frac{1}{2}\right)^{n - 1}\right)^{k - 1}.

c) Expected number of rounds: \displaystyle \frac{2^{n - 1}}{n}.

Step-by-step explanation:

<h3>a)</h3>

To decide the "odd" person, either of the following must happen:

  • There are (n - 1) heads and 1 tail, or
  • There are 1 head and (n - 1) tails.

Assume that the coins here all are all fair. In other words, each has a 50\,\% chance of landing on the head and a

The binomial distribution can model the outcome of n coin-tosses. The chance of getting x heads out of

  • The chance of getting (n - 1) heads (and consequently, 1 tail) would be \displaystyle {n \choose n - 1}\cdot \left(\frac{1}{2}\right)^{n - 1} \cdot \left(\frac{1}{2}\right)^{n - (n - 1)} = n\cdot \left(\frac{1}{2}\right)^n.
  • The chance of getting 1 heads (and consequently, (n - 1) tails) would be \displaystyle {n \choose 1}\cdot \left(\frac{1}{2}\right)^{1} \cdot \left(\frac{1}{2}\right)^{n - 1} = n\cdot \left(\frac{1}{2}\right)^n.

These two events are mutually-exclusive. \displaystyle n\cdot \left(\frac{1}{2}\right)^n + n\cdot \left(\frac{1}{2}\right)^n  = 2\,n \cdot \left(\frac{1}{2}\right)^n = n \cdot \left(\frac{1}{2}\right)^{n - 1} would be the chance that either of them will occur. That's the same as the chance of determining the "odd" person in one round.

<h3>b)</h3>

Since the coins here are all fair, the chance of determining the "odd" person would be \displaystyle n \cdot \left(\frac{1}{2}\right)^{n - 1} in all rounds.

When the chance p of getting a success in each round is the same, the geometric distribution would give the probability of getting the first success (that is, to find the "odd" person) in the kth round: (1 - p)^{k - 1} \cdot p. That's the same as the probability of getting one success after (k - 1) unsuccessful attempts.

In this case, \displaystyle p = n \cdot \left(\frac{1}{2}\right)^{n - 1}. Therefore, the probability of succeeding on round k round would be

\displaystyle \underbrace{\left(1 - n \cdot \left(\frac{1}{2}\right)^{n - 1}\right)^{k - 1}}_{(1 - p)^{k - 1}} \cdot \underbrace{n \cdot \left(\frac{1}{2}\right)^{n - 1}}_{p}.

<h3>c)</h3>

Let p is the chance of success on each round in a geometric distribution. The expected value of that distribution would be \displaystyle \frac{1}{p}.

In this case, since \displaystyle p = n \cdot \left(\frac{1}{2}\right)^{n - 1}, the expected value would be \displaystyle \frac{1}{p} = \frac{1}{\displaystyle n \cdot \left(\frac{1}{2}\right)^{n - 1}}= \frac{2^{n - 1}}{n}.

You might be interested in
Suppose the line of best fit is being found for some data points that have an r-value of 0.657. If the standard deviation of the
oksian1 [2.3K]
We are given a line with the following data:

r-value = 0.657 (r)
standard deviation of x-coordinates = 2.445 (Sx)
standard deviation of y-coordinates = 9.902 (Sy)

We are asked to find the slope of the line up to 3 decimal places.

To find the slope of the line, based on the data that we have, we can use this formula:

slope, b = r * (Sy / Sx) 

substitute the values to the formula:

b = 0.657 * ( 9.902 / 2.445 )

Solve for the b.

Therefore, the slope of the line is 

b = 2.66078, round off to three decimal places:

b = 2.661 is the slope of the line. 
5 0
3 years ago
Read 2 more answers
What is (1.2)to the power of 4
Cloud [144]

Answer:2.0736

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Use the equation
Cerrena [4.2K]

Answer:-  The pressure 'p' will be infinity at volume 'V'=0.

Explanation:-

The given equation :- p=\frac{8.31}{V} where p is a rational function dependent on V.

A rational function is undefined for denominator = 0.

In the given function V is the denominator , if V=0 then the function will be undefined.

⇒ p will approach to zero at V=0.

For assistance see the attachment , the graph of the given function represented by vertical assymptote which approaches to 0 but never touches it..

7 0
3 years ago
Read 2 more answers
What is the radius of the circle?<br>in units<br><br>please help​
Sergio [31]

Answer:

4 units

Step-by-step explanation:

i hope its right

4 0
3 years ago
Read 2 more answers
Aaron covers 300 miles in 6 hours, what's his average speed?​
galina1969 [7]

Answer:

50mph

Step-by-step explanation:

Speed=v

time=t

distance=s

v=s/t

v=300m/6h

v=50mph

Let me know if my answer is wrong.

Hope this helps :) ❤❤❤

6 0
3 years ago
Read 2 more answers
Other questions:
  • HURRRY The volume of a cylinder is given by the formula V=pir2h where r is the radius of the cylinder and h is the height. Which
    9·1 answer
  • 3m2+7=55 answer please
    14·1 answer
  • ONLY ANSWER IF YOU KNOW IT PLS ILL GIVE THE CORRECT BRAINLIEST!
    13·1 answer
  • A coin will be tossed three times, and each toss will be recorded as heads (
    5·1 answer
  • 18.79+2.11+(-1.92)+17.28
    15·1 answer
  • Solve the system of equations using substitution.
    6·1 answer
  • If the angle bisectors of a pair of opposite angles of a quadrilateral are the opposite sides of a parallelogram formed by the t
    15·1 answer
  • The height of a triangle is 5 less than its base. The area of the triangle is 42 square inches. Find the height and base
    15·1 answer
  • Help please now i cant figure it out
    8·1 answer
  • The table of values represents an exponential function f(x).
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!