A counting challenge from Science Buddies
Key concepts Mathematics Probability Chance Strategy Cards Introduction Background Materials Preparation
Procedure
• Decide which type of card you will investigate first. Draw cards from the top of the deck and flip them over one at a time, counting as you go, and stop when you see that type of card. How many cards did you draw until you reached that card? Write down the answer in your table.
• Shuffle the deck again and repeat this process, flipping over the cards and looking for the same type of card. How many cards did it take this time? Write down the answer in your table. Repeat this for a total of 10 times for one type of card. • Repeat this process for each of the four types of cards you picked to investigate. This means that you will have looked for each type of card a total of 10 times. • Calculate the average number of cards you drew to reach each type of card. Label the last row in your table "Averages" and write them in this row.
• Which types of cards were the easiest to draw? Which were the most difficult? How do you think the chances of drawing a card relate to the total number of that card type in the deck?
• Based on what you saw in this activity, how do you think probability can help you choose the right strategy in a card game?
• Extra: A more advanced way of showing the results of your experiment would be to make histograms, which are a type of graph to show distributions. Try making a separate histogram for each type of card you tested by graphing the number of cards drawn for each trial separately in a bar graph. When all of the bars are lined up next to each other, what does the overall shape of the distribution look like?
• Extra: The probability of drawing a particular type of card also depends on the number of cards drawn each time. Try doing this activity again but draw samples of three, five or seven cards at a time. Do your chances improve as more cards are taken?
• Extra: Probabilities can change your strategies for playing a card game. Can you design an experiment to show how probabilities can help you choose cards and win "Go Fish"? What about other popular card games? Can you invent your own game based on probabilities?
Observations and results
Did it take fewer draws to reach a certain color than it took to reach a certain suit or kind of card? Did it take even more draws to reach a specific card? Mathematicians measure probability by counting and using some very basic math, like addition and division. For example, you can add up the number of spades in a complete deck (13) and divide this by the total number of cards in the deck (52) to get the probability of randomly drawing a spade: 13 in 52, or 25 percent. If you were investigating red cards, kings or the queen of hearts, the odds of randomly drawing one of these from a complete deck are 50 percent (26 in 52); about 7.7 percent (four in 52); or about 1.9 percent (one in 52), respectively. This is why, on average (when done over enough trials), it is easier to draw a red card than a spade, a spade than a king, and a king than the queen of hearts. As you draw cards from a deck, the odds of finding your card change. For example, if you are looking for a spade and do not get it on your first draw, there are still 13 spades in the deck but the deck now holds only 51 cards, so your odds of drawing a spade on the second draw are 13 in 51, or about 25.5 percent. This may not seem like much of an improvement, but with every draw the odds continue to increase.More to explore
Unraveling Probability Paradoxes from Scientific American
Calculating the Probability of Simple Events from neoK12
4 Great Math Games from Scholastic, Inc.
Classified Index of Card Games from John McLeod
Pick a Card, Any Card from Science Buddies
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If we take $(c)$ for example,
The question says we know that one of the cards is a spade, what is the probability that both are spades? We need to apply conditional probability.
If Event $A$ is getting both as spades and Event $B$ is getting at least one spade in draw of $2$ cards then
i) $P(B) = \displaystyle \frac{{52 \choose 2} - {39 \choose 2}}{52 \choose 2} = \frac{15}{34}$
Numerator is number of ways to get at least one spade in a draw of $2$ cards, which is all possible draws of $2$ cards minus draws where none of the two cards is a spade.
ii) $ P(A\cap B) = \displaystyle \frac{{13 \choose 2}}{52 \choose 2} = \frac{1}{17}$
[please note that $P(A \cap B) = P(A)$]
So the desired probability $ \displaystyle P (A|B) = \frac{P(A\cap B)}{P(B)} = \frac{2}{15}$.
Now can you use this to evaluate others? If you get stuck, let me know.
While the above can be simplified a bit more, I have gone ahead and used standard working which is important to get used to for conditional probability.