"Three cards are randomly selected from a standard 52-card deck. What is the probability of getting 3 hearts or 3 numbers less than 6 (count aces as 1)?"
I think the answer is 0.0645, but I'm not sure if this is right. Could someone check this?
"Half of a circle is inside a square and half is outside. If a point is selected at random inside the square, find the probability that the point is NOT inside the circle."
I know the area of the square is 4r^2 and the area of the circle is pie r^2, but other than that I have no idea how to solve this. Would someone please help me?
P1=P(3 hearts)=12/52*(11/51)*(10/50)
P2=P(3 numbers each less than 6)
=P(5 or less out of 52)*P(5 or less out of 51)*P(5 or less out of 50)
=(5*4)/52 * (19/51) * (18/50)
Add P1 and P2 to get 4/65.
To calculate the probability of getting 3 hearts or 3 numbers less than 6 from a standard 52-card deck, you need to consider the total number of favorable outcomes and the total number of possible outcomes.
1. Total number of favorable outcomes:
For 3 hearts: There are 13 hearts in a deck, so the probability of drawing a heart three times is (13/52) * (12/51) * (11/50).
For 3 numbers less than 6: There are 4 of each number (2, 3, 4, 5) in a deck, so the probability of drawing a number less than 6 three times is (4/52) * (3/51) * (2/50).
2. Total number of possible outcomes:
The total number of ways to choose 3 cards from a 52-card deck is given by the combination formula, denoted as C(52, 3) = 52! / (3! * (52-3)!), which simplifies to 22,100.
Now, calculate the probability:
P(3 hearts or 3 numbers less than 6) = (favorable outcomes) / (total outcomes)
P(3 hearts or 3 numbers less than 6) = [(13/52) * (12/51) * (11/50)] + [(4/52) * (3/51) * (2/50)]
P(3 hearts or 3 numbers less than 6) ≈ 0.0645
Therefore, your initial answer of approximately 0.0645 is correct.
Now, let's move on to the second question regarding the probability of a point randomly selected inside a square not being inside a circle.
1. Calculate the area of the square: Since the square's side length is r, the area is (2r) * (2r) = 4r^2.
2. Calculate the area of the circle: The circle's diameter is equal to the square's side length (2r), so the radius is r. The area of the circle is πr^2.
3. Find the area of the overlapping regions: The overlapping region consists of two semicircles. The area of one semicircle is (1/2) * πr^2.
4. Subtract the area of the overlapping regions from the square's total area to determine the area of the space outside the circle within the square: 4r^2 - (2 * (1/2) * πr^2) = 4r^2 - πr^2 = (4 - π) r^2.
5. Calculate the probability that a point is NOT inside the circle: The probability is given by (area outside the circle) / (total area of the square) = [(4 - π) r^2] / (4r^2) = (4 - π) / 4.
Therefore, the probability that a point randomly selected inside the square is NOT inside the circle is (4 - π) / 4.
To find the probability of getting 3 hearts or 3 numbers less than 6 when randomly selecting three cards from a standard 52-card deck, we need to calculate the total number of favorable outcomes and the total number of possible outcomes.
For getting 3 hearts:
There are 13 hearts in a deck, so the first card drawn can be any of the 13 hearts.
After drawing the first heart, there are now 51 cards left in the deck, out of which 12 are hearts.
So, the second card drawn must be a heart, which has a probability of 12/51.
Similarly, the third card drawn must also be a heart, which has a probability of 11/50.
Hence, the probability of getting 3 hearts is (13/52) * (12/51) * (11/50) = 0.013255.
For getting 3 numbers less than 6:
There are four 2's, three 3's, and two each of 4's and 5's in a deck, totaling to 11 cards that are numbers less than 6. The first card drawn can be any of these 11 cards.
After drawing the first card, there are now 51 cards left in the deck, out of which there are 10 cards that are numbers less than 6.
So, the second card drawn must be a number less than 6, which has a probability of 10/51.
Similarly, the third card drawn must also be a number less than 6, which has a probability of 9/50.
Hence, the probability of getting 3 numbers less than 6 is (11/52) * (10/51) * (9/50) = 0.012019.
To find the probability of either getting 3 hearts or 3 numbers less than 6, we sum up the individual probabilities because these two events are mutually exclusive (i.e., can't occur simultaneously).
Therefore, the probability of getting 3 hearts or 3 numbers less than 6 is 0.013255 + 0.012019 = 0.025274.
Regarding the second question about the probability of a point being outside the circle when randomly selected inside a square, we can determine it by finding the ratio of the area outside the circle to the total area of the square.
The area of a square is given by side length squared. Let's assume the side length of the square is 's'.
So, the area of the square is s^2.
The area of a circle is given by the formula pi * radius squared. Let's assume the radius of the circle is 'r'.
So, the area of the circle is pi * r^2.
Since the circle is half inside and half outside the square, the radius of the circle must be equal to half the side length of the square (r = s/2).
Substituting the values, the area of the square is s^2 and the area of the circle is (pi * (s/2)^2).
The probability of a point being inside the circle is given by the area of the circle divided by the area of the square.
So, the probability of the point being inside the circle is [(pi * (s/2)^2) / s^2] = (pi/4).
Since the probability of being inside the circle and the probability of being outside the circle must add up to 1, the probability of the point being outside the circle can be calculated by subtracting the probability of being inside the circle from 1.
Therefore, the probability of the point being outside the circle is 1 - (pi/4).
Please note that the exact value of pi is generally approximated as 3.14, so you could further simplify the expression if needed.