If red has a probability of occurrence of .70 and black has a probability of occurrence of .30 and they are the only choices what is the probability that you pick the right color 10 out of 10 times and what is the standard deviation. I keep getting (.70.50) + (.30.50) = .50, IT THIS CORRECT AND WHAT IS STD DEV?

Question

If red has a probability of occurrence of .70 and black has a probability of occurrence of .30 and they are the only choices what is the probability that you pick the right color 10 out of 10 times and what is the standard deviation. I keep getting (.70.50) + (.30.50) = .50, IT THIS CORRECT AND WHAT IS STD DEV?

Answer 1

To calculate the probability of picking the right color 10 out of 10 times, you need to calculate the probability of each individual occurrence and multiply them together. Given that red has a probability of occurrence of 0.70 and black has a probability of occurrence of 0.30, the calculation would be as follows:

Probability of picking the right color (red or black) in one trial:
P(red) = 0.70
P(black) = 0.30

Probability of picking the right color 10 out of 10 times:
P(right color 10 out of 10) = P(red) * P(red) * P(red) * P(red) * P(red) * P(red) * P(red) * P(red) * P(red) * P(red)

Since you are picking the right color every time, the probability remains the same for each trial:

P(right color 10 out of 10) = (0.70) * (0.70) * (0.70) * (0.70) * (0.70) * (0.70) * (0.70) * (0.70) * (0.70) * (0.70)

Simplifying this expression, you get:

P(right color 10 out of 10) = 0.70^10 ≈ 0.028

So, the probability of picking the right color 10 out of 10 times is approximately 0.028, or 2.8%.

Now let's calculate the standard deviation. The standard deviation measures the dispersion or spread of a distribution. In this case, since you only have two possible outcomes (red or black), it is a binomial distribution. The formula to calculate the standard deviation of a binomial distribution is:

Standard Deviation (σ) = √[n * p * (1 - p)]

where:
n = number of trials
p = probability of success in one trial

Using the values from the problem:

Standard Deviation (σ) = √[10 * 0.70 * (1 - 0.70)]

Calculating this expression, you get:

Standard Deviation (σ) = √[2.1]

So, the standard deviation is approximately 1.45.

To summarize:
- The probability of picking the right color 10 out of 10 times is approximately 2.8%.
- The standard deviation is approximately 1.45.