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]
2 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]2 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
Please help me! It's due today at 8:30!!!
klemol [59]

Answer:

137.842

Step-by-step explanation: hope this helps :) can I please get a brain list :)

5 0
2 years ago
Read 2 more answers
Drew invests $70.00. It earns 50% interest annually.
VladimirAG [237]

Step-by-step explanation:

50% interest annually.

that means he gets 50% interest of the invested capital every year.

and that means he gets 50% of $70 in one year.

70 = 100%

1% = 100%/100 = 70/100 = $0.70

50% = 1%×50 = 0.7 × 50 = $35

he will earn $35 interest in one year.

as you noticed: 50% simply means 1/2 (as 100% stands for the whole).

3 0
1 year ago
Find the sum of and 6/7 and 2/3 express your answer in lowest terms.
tensa zangetsu [6.8K]

Answer:

1 11/21

Step-by-step explanation:

6/7 + 2/3 = 32/21

32/21 simplified is 1 11/21

6 0
3 years ago
**Two trains are traveling at constant speeds on different tracks. What is the speed of Train B? *
Eduardwww [97]

Answer:

um what?

Step-by-step explanation:

4 0
2 years ago
alisia goes to the gym every 3 days Luis goes to the gym every 4 days they both are at on the 12th day what is the next day they
jenyasd209 [6]
24th is the answer, you just double the 12 because it’s both a number 3 and 4 go into
6 0
3 years ago
Other questions:
  • Please help! - Kim needs to place 32 mystery books and 56 biography books on library shelves. She is to place the maximum number
    5·2 answers
  • Solve 82×41 , show ur work !
    10·1 answer
  • 99 POINTS AND BRAINLIST!!!! BE FIRST WITH THE "RIGHT" ANSWER!!
    15·2 answers
  • How do you do this???????
    6·1 answer
  • Calculate the size of each of the unknown angles marked on the following diagrams.
    15·1 answer
  • Which expression is equivalent to (x4) -5
    13·1 answer
  • If the length of the radius of the circle is 6, what is the area of the square? 6 72 48 12 144​
    15·1 answer
  • (4 ⅓)C - (8/6)C +12 = 6<br>plz help me
    14·2 answers
  • On Tuesday, Karen finished all her chores in 3/4 of an hour. On Wednesday, she finished her chores in 5/12 of an hour. How much
    10·1 answer
  • 200 people are asked about three brands of beverages Coffee, Tea and Chocolate. 36 like Coffee only, 46 like Coffee but not Tea,
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!