jjjinapinch wrote:Terry holds 12 cards, each of which is red, white, green, or blue. If a person is to select a card randomly from the cards Terry is holding, is the probability less than 1/2 that the card selected will be either red or white?
(1) The probability that the person will select a blue card is 1/3
(2) The probability that the person will select a red card is 1/6
We are given that Terry has 12 total cards. The cards are colored red, white, green, or blue. We need to determine whether the probability of selecting a red or a white card is less than ½.
Remember, since we are determining the probability of selecting a red or a white card, we must add the probabilities.
Is P(red card) + P(white card) < ½?
Since the sum of all probabilities in a sample space is equal to 1, we also know that:
P(red card) + P(white card) + P(blue card) + P(green card) = 1
P(red card) + P(white card) = 1 - [P(blue card) + P(green card)]
Thus, if we can determine the sum of the probabilities of selecting a red card and of selecting a white card OR the sum of the probabilities of selecting a blue card and of selecting a green card, we also could determine the probability of selecting a red or white card.
Statement One Alone:
The probability that the person will select a blue card is 1/3.
Since we don't know the probability of selecting a green card, we cannot determine:
1 - [P(blue card) + P(green card)] OR
P(red card) + P(white card)
For instance, if there were 4 blue, 2 red, 1 white and 5 green balls, the probability of selecting white or red card is (2 + 1)/12 = 3/12 = 1/4. On the other hand, if there were 4 blue, 2 red, 4 white and 2 green balls, the probability of selecting a white or red card is (2 + 4)/12 = 6/12 = 1/2.
Thus, statement one is not sufficient to answer the question.
Statement Two Alone:
The probability that the person will select a red card is 1/6.
Since we don't know the probability of selecting a white card, we cannot determine:
P(red card) + P(white card)
We can take the above example of 4 blue, 2 red, 1 white and 5 green balls versus 4 blue, 2 red, 4 white and 2 green balls (notice that probability of a red card is 2/12 = 1/6 in either case), we see that we get different probabilities for a red or white card.
Thus, statement two is not sufficient to answer the question.
Statements One and Two Together:
Using both statements together we know the following:
P(red card) = 1/6
P(blue card) = 1/3
Substituting this into our two expressions we have:
P(red card) + P(white card) = 1/6 + P(white card) = ?
1 - [1/3 + P(green card)] = ?
We see that we still do not have enough information to determine whether the probability of selecting a red card or a white card is less than ½. In the examples above, notice that probability of selecting a blue card is 4/12 = 1/3 and probability of selecting a red card is 2/12 = 1/6. Those two examples show us that we get different probabilities of selecting a white or red card depending on the number of green and white cards in the deck.
Answer: E
