a)One game in a state lottery requires you to pick 1 heart, 1 club, 1 diamond, and 1 spade, in that order, from the 13 cards in each suit. What is the probability of getting all four picks correct and winning $5000
b)If three of the four selections in part (a) are correct, the player will win $200. What is the probability of this outcome?
The answer provided above is wrong
The probability of all events occurring = product of the individual probabilities.
a) (1/13)^4 = ?
b) (1/13)^3(12/13) = ?
a) To calculate the probability of getting all four picks correct, we need to determine the probability of each individual pick being correct and then multiply them together.
First, let's consider the first pick, which is a heart. Since there are 13 cards in each suit and we need to choose 1 heart, the probability of selecting a heart correctly is 1/13.
Second, we need to pick a club. After selecting a heart correctly, only 12 cards are left in each suit. Therefore, the probability of selecting a club correctly is 1/12.
Third, we need to pick a diamond. After selecting a heart and a club correctly, there are 11 cards left in each suit. Hence, the probability of selecting a diamond correctly is 1/11.
Finally, we need to pick a spade. After selecting a heart, a club, and a diamond correctly, only 10 cards are left in each suit. The probability of selecting a spade correctly is 1/10.
To find the probability of getting all four picks correct, we multiply the probabilities of each individual pick:
(1/13) * (1/12) * (1/11) * (1/10) = 1/17160
Therefore, the probability of winning $5000 by getting all four picks correct is 1 in 17,160.
b) Similarly, to calculate the probability of getting three out of four picks correct, we need to consider all possible combinations of having three correct picks out of four.
There are four ways in which we can have three correct picks: heart, club, diamond (spade incorrect); heart, club, spade (diamond incorrect); heart, diamond, spade (club incorrect); club, diamond, spade (heart incorrect).
For each of these combinations, we can calculate the probability as follows:
1st combination: (1/13) * (1/12) * (1/11) * (10/10) = 1/1716
2nd combination: (1/13) * (1/12) * (10/11) * (1/10) = 1/1716
3rd combination: (1/13) * (10/12) * (1/11) * (1/10) = 1/1716
4th combination: (10/13) * (1/12) * (1/11) * (1/10) = 1/1716
Since there are four possible combinations, we need to sum up the probabilities:
(1/1716) + (1/1716) + (1/1716) + (1/1716) = 4/1716
Therefore, the probability of winning $200 by getting three out of four picks correct is 4 in 1716.
a) To find the probability of getting all four picks correct and winning $5000, we need to calculate the probability of each individual pick being correct and multiply them together.
There are 13 hearts, and we need to pick 1. So the probability of picking the correct heart is 1/13.
After picking a heart, there are 12 clubs left, and we need to pick 1. So the probability of picking the correct club is 1/12.
After picking a heart and a club, there are 11 diamonds left, and we need to pick 1. So the probability of picking the correct diamond is 1/11.
Finally, after picking a heart, a club, and a diamond, there are 10 spades left, and we need to pick 1. So the probability of picking the correct spade is 1/10.
To find the overall probability, we multiply these probabilities together:
P(all picks correct) = (1/13) * (1/12) * (1/11) * (1/10) = 1/17160.
So the probability of getting all four picks correct and winning $5000 is 1/17160.
b) To find the probability of three out of four selections being correct and winning $200, we need to consider the possible combinations of getting three out of the four picks correct.
There are four ways to choose which pick is incorrect. For each incorrect pick, there are 13-1=12 possible incorrect cards in the corresponding suit.
So, the probability of getting exactly three picks correct is:
[P(choose heart correctly) * P(choose club correctly) * P(choose diamond correctly) * P(choose spade incorrectly)] +
[P(choose heart correctly) * P(choose club correctly) * P(choose diamond incorrectly) * P(choose spade correctly)] +
[P(choose heart correctly) * P(choose club incorrectly) * P(choose diamond correctly) * P(choose spade correctly)] +
[P(choose heart incorrectly) * P(choose club correctly) * P(choose diamond correctly) * P(choose spade correctly)]
P(three picks correct) = [(1/13) * (1/12) * (1/11) * (12/10)] +
[(1/13) * (1/12) * (11/11) * (1/10)] +
[(1/13) * (11/12) * (1/11) * (1/10)] +
[(12/13) * (1/12) * (1/11) * (1/10)]
Simplifying, we get:
P(three picks correct) = (12/13) * (1/11) * (1/10) +
(1/13) * (11/12) * (1/11) * (1/10) +
(1/13) * (1/12) * (11/11) * (1/10) +
(1/13) * (1/12) * (1/11) * (12/10)
P(three picks correct) = (12/1430) + (11/1560) + (1/1560) + (12/1716) = 143/17160
So, the probability of getting three out of four selections correct and winning $200 is 143/17160.