Question

a jar contains 22 red marbles number 1 to 22 and 52 blue marbles number 1 to 50 to a marble is drawn at random from the drawer find the probability of the given event around solution to three decimal places

Answer 1

We are given a jar that contains 22 red marble( 1 to 20) and 52 blue marbles (1 to 52). We can proceed to find the solution for each part of the question.

PART 1

Let the probability that the marble is red be P(r).

Therefore,

`P(r)=\frac{Number\text{ of red balls}}{\text{Total number of balls}}`

This gives,

`\begin{gathered} P(r)=(22)/(22+52)=(22)/(74) \\ \therefore P(r)=(11)/(37)=0.297 \end{gathered}`

Therefore, the probability that the marble is red is:

ANSWER= 0.297

PART 2:

Let the probability of picking odd-numbered balls be P(o)

Therefore,

`P\mleft(o\mright)=\frac{Number\text{ of odd balls}}{\text{Total number of balls}}`

We already know that the total number of balls is 72 for the previous question. Therefore, the total number of oddballs will be the sum of odd red balls and odd blue balls. This consists of 11 odd red balls and 26 odd blue balls.

Therefore,

`\begin{gathered} P(o)=(26+11)/(74)=(37)/(74) \\ \therefore P(o)=0.5 \end{gathered}`

The probability of picking odd-numbered balls is

ANSWER = 0.5

PART 3:

Let the probability of picking a red or odd-numbered ball be P(r U o)

`P(r\cup o)=P(r)+p(o)-p(r\cap o)`

Since we already have the values of P(r) and P(o), therefore we only need to find p(r n o).

p(r n o) is the probability of the ball being red and odd. The number of the red and oddball is 11.

Therefore,

`\begin{gathered} P(r\cap o)=\frac{nu\text{mber of red and odd balls}}{\text{Total number of balls}} \\ =(11)/(74) \\ =0.149 \end{gathered}`

This implies that,

`\begin{gathered} P(r\cup o)=P(r)+p(o)-p(r\cap o) \\ P(r\cup o)=0.297+0.5-0.149 \\ \therefore P(r\cup o)=0.648 \end{gathered}`

Hence, the probability of picking a red or odd-numbered ball is

ANSWER = 0.648

PART 4:

Let the probability of picking a blue or even-numbered ball be P(b U e)

Therefore,

`P(b\cup e)=p(b)+p(e)-p(b\cap e)`

From the above formula, we would need to figure out all the parts. p(b) represents the probability of blue marble. This gives,

`\begin{gathered} p(b)=\frac{Number\text{ of blue balls}}{\text{Total number of balls}} \\ \therefore p(b)=(52)/(74)=0.703 \end{gathered}`

p(e) represents the probability of even balls. The total number of even balls will be the sum of the even red balls and even blue balls.

`\begin{gathered} p(e)=(26+11)/(74)=(37)/(74) \\ \therefore p(e)=0.5 \end{gathered}`

p(b n e) represents the probability of blue and even balls. We have 26 blue and even balls

`\begin{gathered} p(b\cap e)=\frac{Number\text{ of blue and even balls}}{\text{Total number of balls}} \\ P(b\cap e)=(26)/(74)=0.351 \end{gathered}`

Therefore,

`\begin{gathered} P(b\cup e)=p(b)+p(e)-p(b\cap e) \\ P(b\cup e)=0.703+0.5-0.351 \\ \therefore P(b\cup e)=0.852 \end{gathered}`

Therefore, the probability of picking a blue or an even ball is:

ANSWER = 0.852