Question

There are five friends. What is the probability that only two of them were born on the same day of the week?​

Answer 1

Final Answer:

The probability that only two of the five friends were born on the same day of the week is 0.1285.

Given:

There are five friends.

To Find:

The probability that only two of them were born on the same day of the week.

Explanation:

The probability of finding the number of successes among repeated trials in an experiment with only two possible outcomes is commonly termed binomial probability.

The repeated trials in an experiment are = n.

The probability of success in an individual trial is = p

The probability of failure in an individual trial is = q =1-p

The number of different combinations of r number of trials selected from the n repeated trials is $=^nC_r$

Then the binomial probability of r number of success as given in  the Bernoulli trial is

$=^nC_r.p^r.q^{(n-r)}\\=^nC_r.p^r.(1-p)^{(n-r)}$.

Step 1 of 3

Apply the binomial probability of the Bernoulli trial after assigning the values to the relevant terms in the formula.

The probability of success is

= The born of a friend on any day out of the seven days of the week

$=p\\=\frac{1}{7}$

The probability of failure in an individual trial is

$=q\\=1-p\\=1-\frac{1}{7}\\=\frac{6}{7}$

Step 2 of 3

The data given in the problem is represented in the following way.

The number of repeated trials is

= The number of total friends

$=n\\=5$

The number of different combinations of r=2 number of friends selected from the five friends is $=^nC_r=^5C_2$

Step 3 of 3

The probability that only two of them were born on the same day of the week

= the binomial probability of 2 number of success as defined in  the Bernoulli trial

$=^5C_2.(\frac{1}{7}) ^2.(\frac{6}{7} )^{(5-2)}\\=\frac{5*4}{2} .(\frac{1}{7}) ^2.(\frac{6}{7} )^3\\=0.1285$

Therefore, the probability that only two of the five friends were born on the same day of the week is 0.1285.

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Answer 2

Answer: Hence, the probability  of only two friends born on the same day is 0.13

Step-by-step explanation:

As it is given in the question that

There are five friends.

and we need to find the probability that only two of them were born on the same day of the week.

-The probability of findind the no. of success among repeated trials in an experiment with only two possible outcomes.

suppose,

Let Success S = { friend is born on a particular day}

Then p = P( S) = 1/7 and so q = 1 — p = 6/7

Here p and q remain constant and so the event of S occurring is a Bernoulli trial.

The probability of x success in n Bernoulli trials is

P(x) = nCx. q ^(n — x ). p ^x

Here n= 5 and x= 2

So P( 2) =5C2 ( 6/7) ^ ( 5 — 2).( 1/7) ^ 2

= 10 (6/7) ^3. (1/7) ^ 2

=0.13

Therefore , the probability that only two of the five were born on the same day of the week is 0.13

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