Question 1194825: This is from my grade 11 foundations math class, not data management,
A card is chosen from a deck of cards, recorded and then replaced. This is done 75 times and a red card from 5 to 9 is chosen 21 times.
a) what is the theoretical probability of a red card between 5 and 9 being chosen?
b) How many times would you expect this event to happen in 75 trials?
C) compare your answers to parts a) and b)
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part A
The set of cards we're focusing on are labeled 5, 6, 7, 8, or 9.
This is 9-5+1 = 5 cards for any given single suit.
The red suits are hearts and diamonds
We have 2 red suits, and 5 cards for each suit, giving 2*5 = 10 red cards that are labeled 5 through 9 inclusive.
The theoretical probability of selecting such a card is 10/52 = 5/26
Answer: 5/26
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Part B
Multiply the theoretical probability found in part A with the number of trials.
(5/26)*75 = 14.423076923077 approximately
This rounds to 14 when rounding to the nearest whole number.
We expect 14 red cards to be between 5 and 9, if 75 identical trials are conducted with replacement. This is a theoretical estimate.
Answer: 14
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Part C
The empirical result was that such a card came up 21 times (stated by your teacher or textbook).
In contrast, the theoretical value found in part B was 14 times.
This difference is simply due to the random nature of selecting cards.
It's like flipping a coin 1000 times. You very likely won't get 500 heads exactly. Instead you might get 487 heads which is fairly close to the theoretical 500.
Going back to the cards, the 14 and 21 aren't too far off.
14/75 = 0.1867 approximately
21/75 = 0.28
Those results aren't spaced out too much either.
The idea is that as the number of trials goes up, the empirical result should get closer to the theoretical value.
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