Fluctuation in the prices of precious metals such as gold have been empirically shown to be well approximated by a normal distribution when observed over short interval of time. In May 1995, the daily price of gold (1 troy ounce) was believed to have a mean of $383 and a standard deviation of $12. A broker, working under these assumptions, wanted to find the probability that the price of gold the next day would be between $394 and $399 per troy ounce. In this eventuality, the broker had an order from a client to sell the gold in the client's portfolio. What is the probability that the client's gold will be sold the next day?
We need to Find the probability where P(394<X<399) in the normal distribution. The ansewr is 0.0885, or about 9%.
Thus, the probability that the Client's gold will be sold the next day is 9%.
To find the probability that the price of gold will be between $394 and $399 per troy ounce the next day, we can use the normal distribution formula.
First, we need to standardize the values using the formula z = (x - μ) / σ, where z is the standard score, x is the value, μ is the mean, and σ is the standard deviation.
For $394 per troy ounce:
z1 = (394 - 383) / 12
z1 = 11 / 12
z1 ≈ 0.92
For $399 per troy ounce:
z2 = (399 - 383) / 12
z2 = 16 / 12
z2 ≈ 1.33
Next, we need to find the probability corresponding to these standardized values using a standard normal distribution table or calculator.
Using the standard normal distribution table, we can find the probabilities corresponding to the z-values:
P(z < 0.92) = 0.8202 (from the table)
P(z < 1.33) = 0.9080 (from the table)
To find the probability between the two values, we subtract the lower probability from the higher probability:
P(0.92 < z < 1.33) = P(z < 1.33) - P(z < 0.92)
P(0.92 < z < 1.33) = 0.9080 - 0.8202
P(0.92 < z < 1.33) ≈ 0.0878
So, the probability that the price of gold will be between $394 and $399 per troy ounce the next day, assuming a normal distribution, is approximately 0.0878 or 8.78%.
Therefore, the probability that the client's gold will be sold the next day is also approximately 0.0878 or 8.78%.
To find the probability that the price of gold will be between $394 and $399 per troy ounce, we need to use the normal distribution.
1. Calculate the z-scores: The z-score formula is given by (X - μ) / σ, where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
For $394:
Z1 = (394 - 383) / 12 = 11 / 12 ≈ 0.917
For $399:
Z2 = (399 - 383) / 12 = 16 / 12 ≈ 1.333
2. Look up the probability: We can use a standard normal distribution table or use a calculator to find the probability associated with each z-score.
The probability associated with Z1 (0.917) is approximately 0.8212.
The probability associated with Z2 (1.333) is approximately 0.9088.
3. Calculate the final probability: Since we want the probability that the price will be between $394 and $399, we need to find the difference between the two probabilities we calculated in step 2.
Probability = P(Z1 ≤ Z ≤ Z2) = P(Z ≤ Z2) - P(Z ≤ Z1)
Probability = 0.9088 - 0.8212 ≈ 0.0876
Therefore, the probability that the client's gold will be sold the next day is approximately 0.0876 or 8.76%.