Question

You are playing a game that uses a deck of cards consisting of 10 green, 10 blue, 10 purple, and 10 red cards. You will

select four cards at random, and you want all four cards to be the same color. You are given two alternatives. You can
randomly select the four cards one at a time, with each card being returned to the deck and the deck being shuffled
before you pick the next card. Alternatively, you can randomly select four cards without the cards being returned to the
deck. Which should you choose? Explain your answer.

Answer 1

For this case the best alternative is 1 since we have a higher probability in order to get the 4 cards with the same color.

See explanation below.

Explanation:

We want on this case to analyze which alternative is better in order to select 4 cards of the sample color.

We have 10 green , 10 blue, 10 purple and 10 red. So in total we have 50 cards.

Alternative 1 : select the four cards one at a time, with each card being returned to the deck and the deck being shuffled before you pick the next card.

Let's assume that we want four blue cards. We need to take in count that this experiment is with replacement. So each time the probability of select on blue card is:

`P(Blue) = (10)/(50)`

And assuming independnet events for each extraction the probability of select the 4 with the same color blue is:

`P(4 Blue) = ((10)/(50))^4 = (1)/(625)=0.0016`

Alternative 2: you can randomly select four cards without the cards being returned to the deck

On this case we assume that the selection is without replacement and for the first extraction we have this:

`P(Blue) = (10)/(50)=(1)/(5)`

For the next extraction since we select one we have this:

`P(2th Blue) = (9)/(49)`

And so on:

`P(3th Blue)= (8)/(48)`

`P(4th Blue)= (7)/(47)`

And the final probability assuming independence would be:

`P(4 Blue)= (1)/(5) * (9)/(49)* (8)/(48)*(7)/(47)=(3)/(3290)=0.0009119`

For this case the best alternative is 1 since we have a higher probability in order to get the 4 cards with the same color.