When messaging a message in Morse code, the chances of a sign being received well are 95%. What is the probability that 50 words of 6 characters will not receive more than 10 wrong words?
i thought because there are 50 words with 6 characters : n=50*6
and p(x>10*6)
To calculate the probability that the 50 words of 6 characters will not receive more than 10 wrong words, you can use the binomial probability formula.
Let's break down the calculation step-by-step:
1. Define the parameters:
- n: The total number of trials or words, which is equal to 50 in this case.
- p: The probability of success or receiving a sign correctly, which is 0.95 (95% chance) according to the given information.
- x: The number of "successes" or correctly received words we want, which is 50 - 10 = 40 (since we want to not receive more than 10 wrong words).
2. Calculate the probability of success (p):
Since the chance of receiving a sign correctly is 95%, the probability of success or p is 0.95.
3. Calculate the probability of failure (q):
The probability of failure or q is simply 1 - p. In this case, q = 1 - 0.95 = 0.05.
4. Use the binomial probability formula to calculate the probability of getting exactly x successes:
P(x) = (nCx) * (p^x) * (q^(n-x))
Applying this to the current scenario:
P(40) = (50C40) * (0.95^40) * (0.05^(50-40))
5. Calculate the required probabilities:
Use a calculator or software to compute the value of P(40).
Keep in mind that P(40) represents the probability of receiving exactly 40 words correctly out of the 50 words.
So, the probability that 50 words of 6 characters will not receive more than 10 wrong words is equal to P(x ≤ 40). To find this probability, you need to sum up the individual probabilities from P(0) to P(40):
P(x ≤ 40) = P(0) + P(1) + P(2) + ... + P(40)
Again, you can use a calculator or software to sum up the probabilities from P(0) to P(40).
I encourage you to use a programming language or software such as Python or Excel to perform these calculations more efficiently.