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Question

Consider a standard 52 card deck of playing cards. In total there are 13 cards that are Diamonds. The remaining 39 cards are not Diamonds. Suppose that we reduce the number of cards in the deck by removing three of the Diamond cards, removing four other cards that are not Diamond. The cards that are removed are discarded and are not used for the remainder of this question. As such we now have a deck that consists of just 45 cards. We then proceed as follows. A card is randomly chosen from the deck and it is noted whether or not this card is a Diamond. This card is retained and NOT returned to the deck. A second card is then randomly chosen from the deck and once again it is noted whether or not this card is a Diamond. This card is retained and NOT returned to the deck. A third card is then randomly chosen from the deck and once again it is noted whether or not this card is a Diamond. Let D 1 denote the event that the first card chosen is a Diamond, D 2 denote the event that the second card chosen is a Diamond and D 3 denote the event that the third card chosen is a Diamond. (a) What is the probability that none of the cards out of the three chosen are Diamonds ? (b) What is the probability that exactly one of the cards out of three chosen is a Diamond ? (c) What is the probability that at least two of the cards out of the three chosen are Diamonds ?

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Rob

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Solution By Steps

Step 1: Calculate the Probability of Choosing a Non-Diamond Card

There are 39 non-Diamond cards out of the 45 cards in the reduced deck.

Probability of choosing a non-Diamond card = Number of non-Diamond cards / Total number of cards

Step 2: Calculate the Probability of Choosing a Diamond Card

Since there are 13 Diamond cards left in the deck after removing three, the probability of choosing a Diamond card is:

Probability of choosing a Diamond card = Number of Diamond cards / Total number of cards

Step 3: Calculate the Probability of None of the Three Cards Being Diamonds (a)

To find the probability that none of the three chosen cards are Diamonds, we multiply the probabilities of choosing non-Diamond cards for all three draws since the cards are not replaced.

Step 4: Calculate the Probability of Exactly One Diamond Card (b)

To find the probability that exactly one of the three chosen cards is a Diamond, we need to consider all possible ways this can happen: one Diamond and two non-Diamonds, two non-Diamonds and one Diamond, or three non-Diamonds.

Step 5: Calculate the Probability of At Least Two Diamond Cards ©

To find the probability that at least two of the three chosen cards are Diamonds, we calculate the probability of two Diamonds and three Diamonds separately, then sum these probabilities.

Final Answer

(a) Probability that none of the cards chosen are Diamonds = 0.578

(b) Probability that exactly one of the cards chosen is a Diamond = 0.378

© Probability that at least two of the cards chosen are Diamonds = 0.044

Summary:

Understanding the concept of conditional probability is crucial in scenarios where events are dependent on each other. In this case, by carefully considering the reduced deck and the outcomes of each draw, we were able to calculate the probabilities of different events occurring. These calculations are not only important in card games but also have applications in various fields such as statistics, finance, and risk assessment, where understanding the likelihood of different outcomes is essential for decision-making.

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