A die is thrown 240 times. What is the probability of getting fifty 2s. plss I have been stuck on this
prob(a 2) = 1/6
prob(not a 2) = 5/6
assuming you want exactly 50 out the 240 to be a 2.
prob(50 2's out of 240 throws)
= C(240,50)(1/6)^50 (5/6)^190
(my calculator can do the two powers but not the
C(240,50) )
To calculate the probability of getting fifty 2s when a die is thrown 240 times, we can use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) is the probability of getting x successes.
n is the total number of trials.
x is the number of successful outcomes.
p is the probability of success on a single trial.
q is the probability of failure on a single trial (which is equal to 1 - p).
nCx represents the combination of n things taken x at a time.
In this case, the number of trials (n) is 240, and we want to find the probability of getting fifty 2s (x = 50).
The probability of getting a 2 on a single die roll is 1/6, as there are six possible outcomes (numbers 1 to 6) and only one of them is a 2 (a successful outcome in this case).
So, we can substitute these values into the formula:
P(50) = (240C50) * (1/6)^50 * (5/6)^(240-50)
Now, we need to calculate the combination (240C50), which represents the number of ways to choose 50 items out of 240 items. This can be calculated using the formula:
(240C50) = (240!)/(50!*(240-50)!)
The exclamation mark represents the factorial of a number, which means multiplying that number by all positive integers less than it down to 1.
Now, let's calculate it step by step:
1. Calculate (240C50):
(240C50) = (240!)/(50!*(240-50)!)
= (240!)/(50!*190!)
2. Simplify (240!):
240! = 240 * 239 * 238 * ... * 2 * 1
3. Simplify (50!):
50! = 50 * 49 * 48 * ... * 2 * 1
4. Simplify (190!):
190! = 190 * 189 * 188 * ... * 2 * 1
5. Substitute these values into the formula:
P(50) = ((240 * 239 * 238 * ... * 2 * 1)/(50 * 49 * 48 * ... * 2 * 1 * 190 * 189 * 188 * ... * 2 * 1)) * (1/6)^50 * (5/6)^(240-50)
6. Now, calculate (1/6)^50 and (5/6)^(240-50):
(1/6)^50 = 1/(6^50)
(5/6)^(240-50) = (5/6)^190
7. Substitute these values back into the formula:
P(50) = (((240 * 239 * 238 * ... * 2 * 1)/(50 * 49 * 48 * ... * 2 * 1 * 190 * 189 * 188 * ... * 2 * 1)) * (1/(6^50)) * ((5/6)^190)
8. Simplify the expression by canceling out common factors in the numerator and denominator. Evaluate the remaining numerical values to get the final probability.
Unfortunately, directly calculating this expression would be computationally intensive due to the large numbers involved. Therefore, it would be more suitable to use statistical software or calculators to get an accurate calculation.
To find the probability of getting fifty 2s when a die is thrown 240 times, we need to consider the total number of outcomes and the number of favorable outcomes.
First, let's determine the total number of outcomes. Since a die has six sides, it can yield any of the numbers from 1 to 6. Therefore, the total number of outcomes when throwing a die 240 times is 6^240 (6 raised to the power of 240).
Next, let's calculate the number of favorable outcomes, which in this case is the number of ways to get exactly fifty 2s when throwing the die 240 times.
This can be calculated using the binomial probability formula:
P(X = k) = (nCk) * (p^k) * (q^(n-k))
Where:
P(X = k) represents the probability of getting exactly k successes,
n represents the total number of trials (in this case, 240 throws),
k represents the number of successful outcomes (in this case, 50 2s),
nCk represents the binomial coefficient (often denoted as "n choose k" and calculated as n! / (k! * (n-k)!)),
p represents the probability of a single success (in this case, the probability of getting a 2 on the die, which is 1/6),
and q is the complement of the probability of a success (in this case, q = 1 - p = 5/6).
Now, let's substitute the given values into the formula:
P(X = 50) = (240C50) * ((1/6)^50) * ((5/6)^(240-50))
To calculate the binomial coefficient (240C50), we need to use the combination formula:
nCk = n! / (k! * (n-k)!)
Substituting the values:
(240C50) = 240! / (50! * (240-50)!)
Use a calculator, statistical software, or online tools to calculate the binomial coefficient (240C50) and evaluate the entire probability formula. This will give you the probability of getting fifty 2s when a die is thrown 240 times.