Question

There are 9 students in a class:2 boys and 7 girls. if the teacher picks 4 at random, what is the probability that everyone in the group is a girl?
(easy)

Answer 1

Answer:

Step 1:

Probability that the first picked student is a girl = Number of girls / Total number of students

=79=79

Step 2:

Probability that the second picked student is a girl = Number of remaining girls / Total number of remaining students

=68=68

Step 3:

Probability that the third picked student is a girl = Number of remaining girls / Total number of remaining students

=57=57

Step 4:

Probability that the fourth picked student is a girl = Number of remaining girls / Total number of remaining students

=46=46

Step 5:

Probability that the all picked student are girls:

=79×68×57×46=518=79×68×57×46=518

Therefore, probability that everyone in the group is a girl =518=518

Step-by-step explanation:

hope u have been understood thanku

Answer 2

$\large\underline{\sf{Solution-}}$

Given that there are 9 students in a class, out of which 2 are boys and 7 are girls.

Teacher want to pick 4 students at random.

We know,

The number of ways in which 'r' objects can be chosen from 'n' distinct objects is

$\boxed{ \bf{ \: ^nC_r \: = \: \frac{n!}{r! \: ( \: n \: - \: r \: )!} \: }}$

So,

Number of ways to choose 4 students out of 9 is

$\rm \:  =  \:  \: ^9C_4$

$\rm \:  =  \:  \: \dfrac{9!}{4! \: (9 - 4)!}$

$\rm \:  =  \:  \: \dfrac{9!}{4! \: 5!}$

$\rm \:  =  \:  \: \dfrac{9 \times 8 \times 7 \times 6 \times 5!}{4 \times 3 \times 2 \times 1 \times \: 5!}$

$\rm \:  =  \:  \: 126$

Now,

Teacher want to pick 4 girls.

So,

Number of ways in which 4 girls can be chosen from 7 girls is

$\rm \:  =  \:  \: ^7C_4$

$\rm \:  =  \:  \: \dfrac{7!}{4! \: (7 - 4)!}$

$\rm \:  =  \:  \: \dfrac{7!}{4! \: 3!}$

$\rm \:  =  \:  \: \dfrac{7 \times 6 \times 5 \times 4!}{4! \times \: 3 \times 2 \times 1}$

$\rm \:  =  \:  \: 35$

Now,

The required probability that everyone in a group is a girl is

$\rm \:  =  \:  \: \dfrac{35}{126}$

Additional Information :-

$\boxed{ \rm{ P(A \: \cup \: B) = P(A) + P(B) - P(A \: \cap \: B)}}$

$\boxed{ \rm{ P(A \: \cap \: B') = P(A) - P(A \: \cap \: B)}}$

$\boxed{ \rm{ P(A '\: \cap \: B') =1 - P(A \: \cup \: B)}}$

$\boxed{ \rm{ P(A '\: \cup \: B') =1 - P(A \: \cap \: B)}}$

$\boxed{ \rm{ P(A) + P(A') = 1}}$