dale placed six cARDS IN A BAG.QONE CARD IS LABELD 1,TWO CARDS ARE LABELD2,AND RTHREE CARDS ARE LABELD 3.FIND THE PROBABILITY OOF PICKING TWO CARDS IN THE ORDER OF THE NUBERS LISTED TH EFIRST CARD IS RETURNED BEFORE THE SECOND IS PICKED ..... P,(1,2)
P(3,NOT1)
p(NOT2,2)
P(1,3)
P(2,3)
P(2,NOT2)
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To find the probabilities of picking the cards in a specific order, we need to know the total number of cards in the bag and the number of cards with each label.
In this case, Dale placed six cards in a bag, with the labels 1, 2, and 3. There is one card labeled 1, two cards labeled 2, and three cards labeled 3.
Now, let's calculate the probabilities for each case:
1. P(1,2): The probability of picking the card labeled 1 first and then the card labeled 2.
To calculate this probability, we need to determine the number of favorable outcomes (ways of picking 1 first and 2 second) and the total number of possible outcomes (all possible combinations of picking two cards).
Favorable outcomes: There is only one card labeled 1, so the number of ways of picking it first is 1. There are two cards labeled 2, so the number of ways of picking one of them second is 2.
Total possible outcomes: There are six cards in total, so the total number of possible combinations of picking two cards is 6C2 = 15 (using the formula for combinations).
Therefore, P(1,2) = (favorable outcomes) / (total possible outcomes) = 1/15.
2. P(3, NOT 1): The probability of picking any card labeled 3 first and then any card that is not labeled 1 second.
To calculate this probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Favorable outcomes: There are three cards labeled 3, so the number of ways of picking one of them first is 3. Since we do not want to pick a card labeled 1 second, there are five cards remaining (6 total cards - 1 card labeled 1).
Total possible outcomes: There are still six cards in total, so the total number of possible combinations of picking two cards is still 6C2 = 15.
Therefore, P(3, NOT 1) = (favorable outcomes) / (total possible outcomes) = (3 * 5) / 15 = 15/15 = 1.
3. P(NOT 2, 2): The probability of picking any card that is not labeled 2 first and then the card labeled 2 second.
To calculate this probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Favorable outcomes: There are four cards that are not labeled 2, so the number of ways of picking one of them first is 4. There are still two cards labeled 2 remaining.
Total possible outcomes: There are still six cards in total, so the total number of possible combinations of picking two cards is still 6C2 = 15.
Therefore, P(NOT 2, 2) = (favorable outcomes) / (total possible outcomes) = (4 * 2) / 15 = 8/15.
4. P(1, 3): The probability of picking the card labeled 1 first and then the card labeled 3 second.
Favorable outcomes: There is only one card labeled 1, so the number of ways of picking it first is 1. There are three cards labeled 3, so the number of ways of picking one of them second is 3.
Total possible outcomes: Still 6C2 = 15.
Therefore, P(1, 3) = (favorable outcomes) / (total possible outcomes) = 1/15.
5. P(2, 3): The probability of picking the card labeled 2 first and then the card labeled 3 second.
Favorable outcomes: There are two cards labeled 2, so the number of ways of picking one of them first is 2. There are three cards labeled 3, so the number of ways of picking one of them second is 3.
Total possible outcomes: Still 6C2 = 15.
Therefore, P(2, 3) = (favorable outcomes) / (total possible outcomes) = (2 * 3) / 15 = 6/15.
6. P(2, NOT 2): The probability of picking the card labeled 2 first and then any card that is not labeled 2 second.
Favorable outcomes: There are two cards labeled 2, so the number of ways of picking one of them first is 2. There are still four cards (6 total cards - 2 cards labeled 2) remaining.
Total possible outcomes: Still 6C2 = 15.
Therefore, P(2, NOT 2) = (favorable outcomes) / (total possible outcomes) = (2 * 4) / 15 = 8/15.
So, the probabilities are as follows:
P(1,2) = 1/15
P(3,NOT1) = 1
P(NOT2,2) = 8/15
P(1,3) = 1/15
P(2,3) = 6/15
P(2,NOT2) = 8/15.