Outside a home, there is a 9-key keypad numbered 1 through 9 . The correct four-digit code will open the garage door. The numbers can be repeated in the code.
(a) How many codes are possible?
(b) What is the probability of entering the correct code on the first try, assuming that the owner doesn’t remember the code?
(a) The number of possible codes is .
(Type an integer or fraction. Simplify your answer.)
(b) The probability that the correct code is given on the first try, assuming that the owner doesn’t remember it is
(Type an integer or fraction. Simplify your answer.)

Question

Outside a home, there is a 9-key keypad numbered 1 through 9 . The correct four-digit code will open the garage door. The numbers can be repeated in the code.
(a) How many codes are possible?
(b) What is the probability of entering the correct code on the first try, assuming that the owner doesn’t remember the code?
(a) The number of possible codes is

.
(Type an integer or fraction. Simplify your answer.)
(b) The probability that the correct code is given on the first try, assuming that the owner doesn’t remember it is
(Type an integer or fraction. Simplify your answer.)

UpStudy AI · Accepted Answer

**Step 1. Count the total number of possible codes**

Since the keypad has $$9$$ digits and each digit in the four-digit code may be repeated, the number of codes is calculated by multiplying the number of choices for each digit:
$$
9 \times 9 \times 9 \times 9 = 9^4 = 6561.
$$
Thus, the number of possible codes is $$6561$$.

**Step 2. Calculate the probability of entering the correct code on the first try**

Since only one code is correct, the probability of choosing the correct code on the first attempt is:
$$
\frac{1}{6561}.
$$

**Final Answers:**

(a) The number of possible codes is $$6561$$.

(b) The probability is $$\frac{1}{6561}$$.
(a) There are 6561 possible codes. (b) The probability of entering the correct code on the first try is $$\frac{1}{6561}$$.

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