Keno is a casino game in which the player has a card with the numbers 1 through 80 on it. The player selects a set of k numbers from the card, where k can range from one to fifteen. The "caller" announces twenty winning numbers, chosen at random from the eighty. the amount won depends on how many of the called numbers match those the player chose. Suppose the player picks ten numbers. What is the probability that among those ten are six winning numbers?
.011479
Well, let's give those numbers a little "keno-nudge"!
The probability of winning depends on the total number of ways you can choose 10 numbers from the 80 available and how many of those numbers are among the 20 winning numbers selected randomly.
To figure out the probability, we need to use a little math magic. First, we need to calculate the number of ways the player can choose 10 numbers from the set of 80. This can be done using the combination formula, denoted by C(n, r), which represents "n choose r".
In this case, n = 80 (total numbers available) and r = 10 (numbers chosen by the player). So, we have C(80, 10).
Next, we need to determine how many ways we can choose exactly 6 winning numbers from the 20 randomly selected numbers. This can be calculated using the combination formula again, with n = 20 and r = 6.
So, we have C(20, 6).
Now, to calculate the probability, we need to divide the number of favorable outcomes (choosing 6 winning numbers) by the total number of possible outcomes (choosing any 10 numbers).
The probability can be calculated as:
P(6 winning numbers) = [C(20, 6) * C(60, 4)] / C(80, 10)
Now, I could crunch all these numbers for you, but let's just say it's pretty slim. So, good luck! And remember, even if the probability is low, you never know when luck might decide to dance its way into your keno game! Keep those numbers crossed!
To calculate the probability of having exactly six winning numbers out of ten picked numbers in the game of Keno, we can use the concept of combinations.
First, we need to determine the total number of possible combinations for selecting 10 numbers out of 80. This can be calculated using the formula for combinations:
C(n, k) = n! / (k! * (n - k)!)
Where:
- n is the total number of items to choose from (in this case, 80)
- k is the number of items to be chosen (in this case, 10)
- ! represents the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1)
Substituting the values into the formula:
C(80, 10) = 80! / (10! * (80 - 10)!)
Next, we need to determine the number of ways to choose exactly six winning numbers out of the ten picked numbers. This can also be calculated using the combination formula:
C(k, x) = k! / (x! * (k - x)!)
Where:
- k is the number of items to choose from (in this case, 10)
- x is the number of items to be chosen (in this case, 6)
Substituting the values into the formula:
C(10, 6) = 10! / (6! * (10 - 6)!)
Finally, to calculate the probability, we divide the number of ways to have exactly six winning numbers by the total number of possible combinations:
Probability = C(10, 6) / C(80, 10)
Therefore, the probability of having exactly six winning numbers out of ten picked numbers in Keno is:
Probability = (10! / (6! * (10 - 6)!) ) / ( 80! / (10! * (80 - 10)! ) )
To find the probability of picking exactly six winning numbers out of ten, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Total Number of Possible Outcomes:
In Keno, the player selects 10 numbers out of the 80 available. The total number of ways to choose 10 numbers from 80 is given by the combination formula: C(80,10). This can be calculated as:
C(80,10) = 80! / (10! * (80-10)!)
Number of Favorable Outcomes:
To have exactly six winning numbers, we need to choose 6 numbers correctly from the 20 winning numbers called and 4 numbers incorrectly from the remaining 60 numbers not called. The number of favorable outcomes can be calculated using the combination formula as well:
C(20,6) * C(60,4) = (20! / (6! * (20-6)!) * (60! / (4! * (60-4)!)
Probability:
The probability of having exactly six winning numbers out of ten can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(6 winning numbers) = (C(20,6) * C(60,4)) / C(80,10)
Now we can substitute the values and calculate the probability.