Question

A bag contains 25 balls numered from 1to 25 a card is drwan at random from the bag find the probability that the number on the drwan card 1)card is divisible by 3or 5 2)a perfect sq number

Answer 1

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$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,
A bag contains 25 balls numbered from 1 to 25
A ball is randomly drawn from the bag

Let S be the Sample space

Let $E_{1}$ be the Event that the drawn ball has a number which is divisible by 3 or 5

Let $E_{2}$ be the Event that drawn ball has a number which is a perfect square

$P(E_{1}) = ?$

$P(E_{2}) = ?$

$\underline{\large{\textbf{Step-1:}}}$
Find Total No. of Outcomes when a ball is drawn from the bag

Total No. of outcomes = Total No. of ways of drawing a ball from 25 Balls

$\implies$ Total No. of Outcomes, $n(S)$ = 25

$\underline{\large{\textbf{Step-2:}}}$
Find No. of Outcomes favoring occurrence of Event '$E_{1}$'

No. of Favorable Outcomes for occurrence of Event '$E_{1}$' = (No. of Multiples of 3 from 1 to 25 )+ (No. of Multiples of 5 from 1 to 25) - (Common Multiples of 3 and 5)

No. of Multiples of 3 from 1 to 25 = quotient of (25 ÷ 3) = 8
(which are 3, 6, 9, 12, 15, 18, 21, 24)

No. of Multiples of 5 from 1 to 25 = quotient of (25 ÷ 5) = 5
(which are 5, 10, 15, 20, 25)

No. of Common Multiples of 3 and 5 = 1
(which is 15)

Substituting Values
No. of Favorable Outcomes for $E_{2}$, $n(E_{2})$ = 8+5-1 = 12

[ $E_{1}$= {3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25} ]

$\underline{\large{\textbf{Step-3:}}}$
Find No. of Outcomes favoring occurrence of Event '$E_{2}$'

No. of Favorable Outcomes for occurrence of Event '$E_{2}$' = Perfect Squares from 1 to 25

*** Perfect Squares are squares of the integers

No. of Favorable Outcomes for $E_{2}$ $n(E_{2})$ = 4
[ $E_{2}$= {4, 9, 16, 25} ]

$\underline{\large{\textbf{Step-4:}}}$
Find No. of Probability of occurrence of Event '$E_{1}$' and '$E_{2}$'

$Probability = \dfrac{No.\: of\: Favorable\: outcomes\: of\: Event}{Total\: No.\: of\: Outcomes\: of\: Experiment}$

$\star$ $P(E_{1}) = \dfrac{n(E_{1})}{n(S)}$

$\implies P(E_{1}) = \dfrac{12}{25}$

$\star$ $P(E_{2}) = \dfrac{n(E_{2})}{n(S)}$

$\implies P(E_{2}) = \dfrac{4}{25}$

Therefore,
✓ Probability that number on the drawn ball is divisible by 3 or 5 = $\dfrac{12}{25}$

✓ Probability that number on the drawn ball is a perfect square = $\dfrac{4}{25}$
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Answer 2

Answer:

Given,

A bag contains 25 balls numbered from 1 to 25

A ball is randomly drawn from the bag

Let S be the Sample space

Let E_{1}E

1

be the Event that the drawn ball has a number which is divisible by 3 or 5

Let E_{2}E

2

be the Event that drawn ball has a number which is a perfect square

P(E_{1}) = ?P(E

1

)=?

P(E_{2}) = ?P(E

2

)=?

\underline{\large{\textbf{Step-1:}}}

Step-1:

Find Total No. of Outcomes when a ball is drawn from the bag

Total No. of outcomes = Total No. of ways of drawing a ball from 25 Balls

\implies⟹ Total No. of Outcomes, n(S)n(S) = 25

\underline{\large{\textbf{Step-2:}}}

Step-2:

Find No. of Outcomes favoring occurrence of Event 'E_{1}E

1

'

No. of Favorable Outcomes for occurrence of Event 'E_{1}E

1

' = (No. of Multiples of 3 from 1 to 25 )+ (No. of Multiples of 5 from 1 to 25) - (Common Multiples of 3 and 5)

No. of Multiples of 3 from 1 to 25 = quotient of (25 ÷ 3) = 8

(which are 3, 6, 9, 12, 15, 18, 21, 24)

No. of Multiples of 5 from 1 to 25 = quotient of (25 ÷ 5) = 5

(which are 5, 10, 15, 20, 25)

No. of Common Multiples of 3 and 5 = 1

(which is 15)

Substituting Values

No. of Favorable Outcomes for E_{2}E

2

, n(E_{2})n(E

2

) = 8+5-1 = 12

[ E_{1}E

1

= {3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25} ]

\underline{\large{\textbf{Step-3:}}}

Step-3:

Find No. of Outcomes favoring occurrence of Event 'E_{2}E

2

'

No. of Favorable Outcomes for occurrence of Event 'E_{2}E

2

' = Perfect Squares from 1 to 25

*** Perfect Squares are squares of the integers

No. of Favorable Outcomes for E_{2}E

2

n(E_{2})n(E

2

) = 4

[ E_{2}E

2

= {4, 9, 16, 25} ]

\underline{\large{\textbf{Step-4:}}}

Step-4:

Find No. of Probability of occurrence of Event 'E_{1}E

1

' and 'E_{2}E

2

'

Probability = \dfrac{No.\: of\: Favorable\: outcomes\: of\: Event}{Total\: No.\: of\: Outcomes\: of\: Experiment}Probability=

TotalNo.ofOutcomesofExperiment

No.ofFavorableoutcomesofEvent

\star⋆ P(E_{1}) = \dfrac{n(E_{1})}{n(S)}P(E

1

)=

n(S)

n(E

1

)

\implies P(E_{1}) = \dfrac{12}{25}⟹P(E

1

)=

25

12

\star⋆ P(E_{2}) = \dfrac{n(E_{2})}{n(S)}P(E

2

)=

n(S)

n(E

2

)

\implies P(E_{2}) = \dfrac{4}{25}⟹P(E

2

)=

25

4

Therefore,

✓ Probability that number on the drawn ball is divisible by 3 or 5 = \dfrac{12}{25}

25

12

✓ Probability that number on the drawn ball is a perfect square = \dfrac{4}{25}

25

4