Question

Please can you answer with full working

Will give brainliest

Answer 1

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

Given that

Laura put green counters and yellow counters in an empty box in the ratio 1 : 3.

Let assume that

Number of green counters = x

Number of yellow counters = 3x

So, Total number of counters = 4x

Now, Probability of getting 2 green counters is

$\rm \: = \: \dfrac{^{x}C_{2}}{^{4x}C_{2}}$

We know,

$\boxed{\tt{ \: ^{n}C_{2} = \frac{n(n - 1)}{2} \: }}$

So, using this identity, we get

$\rm \: = \: \dfrac{\dfrac{x(x - 1)}{2} }{\dfrac{4x(4x - 1)}{2} }$

$\rm \: = \: \dfrac{x - 1}{4(4x - 1)}$

Now, Further given that

$\rm \: Probability \: of \: getting \: 2 \: green \: counter = \dfrac{5}{84}$

So, from above two equations, we concluded that

$\rm \: \dfrac{x - 1}{4(4x - 1)} = \dfrac{5}{84}$

$\rm \: \dfrac{x - 1}{4x - 1} = \dfrac{5}{21}$

$\rm \: 21x - 21 = 20x - 5$

$\rm \: 21x - 20x = 21 - 5$

$\rm\implies \:x = 16$

Hence,

Number of green counters = 16

Number of yellow counters = 3 × 16 = 48

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ADDITIONAL INFORMATION

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

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

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

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

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

Answer 2

Refer the Given Attachments