(a) How can we convert the information on credit card defaults into a complete probability distribution using Excel?
(b) What is the probability that none of the credit card holders will default?
(c) What is the probability that at least one card holder will default?
(d) How many of the new credit card holders would we expect to default?
(e) What is the expected average variation around the expected number of defaults?
The probability distribution shows the chances of different numbers of credit card defaults, with an expected number of 0.84 defaults and a standard deviation of approximately 0.986.
Probability distribution:
- P(X = 0) = 0.330014
- P(X = 1) = 0.377355
- P(X = 2) = 0.226413
- P(X = 3) = 0.089153
- P(X = 4) = 0.025040
- P(X = 5) = 0.004807
- P(X = 6) = 0.000710
- P(X = 7) = 0.000083
- P(X = 8) = 0.000008
- P(X = 9) = 0.000001
- P(X = 10) = 0.000000
- P(X = 11) = 0.000000
- P(X = 12) = 0.000000
Probability of none of the card holders defaulting:
P(X = 0) = 0.330014
Probability of at least one card holder defaulting:
P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.330014 = 0.669986
Expected number of defaults:
Expected number of defaults = n * p = 12 * 0.07 = 0.84
Expected average variation around the expected number of defaults:
Standard deviation = √(n * p * (1 - p)) = √(12 * 0.07 * (1 - 0.07)) ≈ 0.986
Probability Distribution Calculation using Excel
To convert the information on credit card defaults into a complete probability distribution using Excel, you can create a table with the number of defaults (X) and their corresponding probabilities. Each row in the table represents a possible number of defaults, and the probability is calculated based on the given data.
Probability of None of the Credit Card Holders Defaulting
The probability of none of the credit card holders defaulting is 0.330014, which means there is a 33.0014% chance that all 12 card holders will not default.
Probability of At Least One Card Holder Defaulting
The probability of at least one card holder defaulting is calculated as 1 - P(X = 0) = 1 - 0.330014 = 0.669986. This means there is a 66.9986% chance that at least one card holder will default.
Expected Number of Defaults
Based on the information provided, the expected number of defaults is calculated as 12 * 0.07 = 0.84. Therefore, we can expect approximately 0.84 of the new credit card holders to default.
Expected Average Variation around the Expected Number of Defaults
The standard deviation around the expected number of defaults is approximately 0.986. This value indicates the average variation or spread of the actual number of defaults from the expected value of 0.84.
In conclusion, understanding the probability distribution and calculations related to credit card defaults helps in managing the risk associated with credit card holders defaulting.