Problem Setup
We need to find the probability when choosing 3 letters from {A, B, C, D, E, F} (6 letters) randomly and with replacement. The goal is to determine the probability that the chosen sequence forms either the word $BAD$ or the word $CAD$.
Total Possible Outcomes
Since the letters are chosen with replacement, each of the 3 positions in the sequence has 6 independent choices (A, B, C, D, E, F).
The total number of possible ordered sequences is calculated as:
$ \text{Total Outcomes} = 6 \times 6 \times 6 = 6^3 = 216 $
Favorable Outcomes
The question requires the probability of forming either $BAD$ or $CAD$. Given the options, the interpretation is that the selected sequence must be one of the permutations of the letters in $BAD$ or one of the permutations of the letters in $CAD$.
1. Permutations related to $BAD$:
The specific letters required are B, A, and D. The distinct permutations of these three letters are:
- $BAD$
- $BDA$
- $ABD$
- $ADB$
- $DAB$
- $DBA$
There are $3! = 6$ such sequences.
2. Permutations related to $CAD$:
The specific letters required are C, A, and D. The distinct permutations of these three letters are:
- $CAD$
- $CDA$
- $ACD$
- $ADC$
- $DAC$
- $DCA$
There are $3! = 6$ such sequences.
These two sets of sequences (permutations of $BAD$ and permutations of $CAD$) are mutually exclusive.
Total Favorable Outcomes = (Number of permutations of $BAD$) + (Number of permutations of $CAD$) = $6 + 6 = 12$.
Calculating the Probability
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
$ \text{Probability} = \frac{\text{Total Favorable Outcomes}}{\text{Total Possible Outcomes}} $
$ \text{Probability} = \frac{12}{216} $
This result matches Option D.
Important Questions from Discrete Probability
- A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is
- Let $X$ and $Y$ be independent Poisson random variables with means $4$ and $2$, respectively. Then, which of the following statements are true?
- Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate $3$ and successive service times are independent exponential random variables having mean $\frac{1}{9}$. Let $P_n$ be the long run probability that there are exactly $n$ customers in the system. Then, which of the following statements are true?
- Let $X$ be a Binomial$(n, p)$ random variable, where $n \in \{5,6\}$ and $p\in \{\frac{1}{4}, \frac{3}{4}\}$. If $X = 3$ is observed, then the maximum likelihood estimate of $(n, p)$ is
- Suppose two fair dice are thrown independently at random. Let $X$ and $Y$ be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
