Each person that works at a company is given a 5-digit code followed by a letter,either uppercase or lowercase.These employees must enter their codes on a keypad to enter and exit the office building The company has 130 employees.
a) How many codes are possible if there are no restrictions?
b) What is the probability of someone entering a code at random and gaining entry to building?
there are 10^5 * 52 unique codes
there are 130 valid ones for access
so, it should be clear what p(access) is for random entry. Not high.
Thank you but I don't understand why is 52 in the first question ;and what is exactly the answer to the question b)
a) To calculate the number of possible codes, we need to consider both the 5-digit number and the letter following it.
For the 5-digit number, there are 10 possible digits (0-9) for each position. Since the code is 5 digits long, there are 10 options for the first digit, 10 options for the second digit, and so on. Thus, there are 10^5 = 100,000 possible 5-digit codes.
For the letter following the 5-digit number, there are 26 possible uppercase letters and 26 possible lowercase letters. So, there are 26 + 26 = 52 possible letters.
To calculate the total number of possible codes, we multiply the number of possible 5-digit numbers by the number of possible letters:
Total number of possible codes = (10^5) * 52 = 5,200,000
Therefore, there are 5,200,000 possible codes if there are no restrictions.
b) To calculate the probability of someone entering the code at random and gaining entry to the building, we need to determine the number of favorable outcomes (codes that grant entry) and the total number of possible outcomes.
Since there are no restrictions on the codes, all 5,200,000 possible codes are considered.
The probability of randomly choosing a code that grants entry to the building is:
Probability = Favorable Outcomes / Total Outcomes
Since there is no information provided on the number of codes that grant entry, we assume it is the same as the total number of possible codes (5,200,000). Therefore, the probability would be:
Probability = 5,200,000 / 5,200,000 = 1
Hence, the probability of someone entering a code at random and gaining entry to the building is 1, which means it is certain they will gain entry.
a) To find the number of possible codes, we need to consider the total number of digits (5) and the number of possible values for each digit.
Since there are 26 lowercase letters, 26 uppercase letters, and 10 digits (0-9), the total number of possible values for each digit is 26+26+10 = 62.
Therefore, the number of possible codes is calculated by raising the number of possible values for each digit (62) to the power of the number of digits (5):
Number of possible codes = 62^5 = 916,132,832
So, there are 916,132,832 possible codes if there are no restrictions.
b) To find the probability of someone entering a code at random and gaining entry to a building, we need to consider the number of successful outcomes (codes that allow entry) and the total number of possible outcomes (all codes).
The number of successful outcomes is one since only one correct code grants entry, while the total number of possible outcomes is equal to the number of possible codes we calculated previously, which is 916,132,832.
So, the probability of gaining entry by entering a code at random is:
Probability = Number of successful outcomes / Total number of possible outcomes
Probability = 1 / 916,132,832 ≈ 1.09 x 10^-9
Therefore, the probability of someone entering a code at random and gaining entry to the building is approximately 1.09 x 10^-9.