Question

4. A box contains 5 green pencils and 7 yellow pencils. Two pencils are chosen at random from the box without replacement. What is the probability they are different colors?

Answer 1

Answer:

To find the probability of two events happening, we must multiply the probability of each separate event happening by the other.

So, for the first time that you pick a pencil, we have to find the probability that it is a yellow pencil. 

Probability = favorable outcomes / total possible outcomes = 7 yellow / 7 yellow + 5 green = 7 yellow pencils / 12 total pencils

Thus, the probability of the first event is 7/12.

According to the question, we want to pick a yellow pencil both times, but we don't replace it.  So, assuming we pick a yellow pencil, we have to subtract that from the next calculation.

Thus, probability of the second event = probable outcomes/ total outcomes = 6 yellow pencils / 6 yellow + 5 green pencils = 6 / 11

Therefore, the second event's probability is 6/11.

To find the total probability, we must multiply the two fractions together.

7 / 12 * 6 / 11 = 42 / 132

But, we want to simplify this fraction, so:

42 / 132 = 42 ÷ 6 / 132 ÷ 6 = 7/22

Thus, the probability is 7/22, or approximately 32%.

Hope this helps you....

Answer 2

Concept:

To solve this answer let's recall the definition of probability.

Probability is the concept that numerically measures the degree of uncertainty and therefore certainty of the occurrence of an event.

Formula required:

P(of occurrence of the event)=no. of outcomes favourable to the occurrence of E/Total possible outcomes.

P(of occurrence of the event)=n(E)/n(S)

Given:

We are given a box with 5 green pencils and 7 yellow pencils and we are asked to pick two pencils randomly without replacing the previous.

To find:

We are asked to find the probability of picking different colour pencils.

Solution:

Let S be the sample space.

Therefore,

n(S)=no. of green pencils+ no. of yellow pencils

n(S)=5+7

n(S) =12

Let G= event of picking a green pencil

Let Y=event of picking a yellow pencil

Therefore,

n(G)=no. of green pencils in the box

n(G)=5

And,

n(Y)=no. of yellow pencils in the box

n(Y)=7

Now, We will see the probability of picking a green pencil

P(G)=n(G)/n(S) (using the formula of probability)

P(G)=5/12

Thus, the probability of the first event is 5/12

As the green pencil is picked, it won't be replaced back. So, the total number of pencils in the box from which we can choose the second pencil is 11. So, now n(S) is updated to 11.

Then, the probability of picking a yellow pencil is

P(Y)=n(Y)/n(S)

P(Y)=7/11

Thus, the probability of the second event is: 7/11

We are asked for the probability of different colours, for that we multiply the first event with the second as they are independent of each other.

Hence, the required probability:

P(picking different colours)=P(G) x P(Y)

P(picking different colours)=5/12 x 7/11

P(picking different colours)=35/132

P(picking different colours)=0.2651