Let's tackle this probability problem step by step. Given that the probability of guessing correctly on any question is 30% (or 0.3), we can use the binomial probability formula to calculate the probabilities for each part of the problem. The binomial probability formula is:
[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} ]
where:
( P(X = k) ) is the probability of getting exactly ( k ) correct answers,
( \binom{n}{k} ) is the binomial coefficient, which represents the number of ways to choose ( k ) successes out of ( n ) trials,
( n ) is the number of trials (in our case, 6 questions),
( k ) is the number of successful outcomes (correct answers),
( p ) is the probability of success on a single trial (30% or 0.3),
( (1-p) ) is the probability of failure on a single trial (70% or 0.7).
Let's calculate each part:
a) Probability of getting more than four correct answers
This means getting either 5 or 6 correct answers.
For 5 correct answers:
[ P(X = 5) = \binom{6}{5} \cdot (0.3)^5 \cdot (0.7)^1 ]
For 6 correct answers:
[ P(X = 6) = \binom{6}{6} \cdot (0.3)^6 \cdot (0.7)^0 ]
The total probability of getting more than four correct answers is the sum of these two probabilities:
[ P(X > 4) = P(X = 5) + P(X = 6) ]
b) Probability of getting at least two correct answers
This means getting 2, 3, 4, 5, or 6 correct answers.
We can calculate the probabilities for each of these cases and sum them up, but it's easier to calculate the probability of getting 0 or 1 correct answers and subtract that from 1.
[ P(X \geq 2) = 1 - [P(X = 0) + P(X = 1)] ]
c) Probability of getting at most three correct answers
This means getting 0, 1, 2, or 3 correct answers.
We sum the probabilities of these four cases:
[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) ]
d) Probability of getting less than five correct answers
This is very similar to part a), but we want the probability of 0, 1, 2, 3, or 4 correct answers.
[ P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) ]
Now, let's calculate each of these probabilities using the binomial formula. I'll start with part a):
For 5 correct answers:
[ P(X = 5) = \binom{6}{5} \cdot (0.3)^5 \cdot (0.7)^1 = 6 \cdot (0.3)^5 \cdot (0.7) ]
For 6 correct answers:
[ P(X = 6) = \binom{6}{6} \cdot (0.3)^6 \cdot (0.7)^0 = 1 \cdot (0.3)^6 ]
Add the two probabilities together to get ( P(X > 4) ).
You can follow these steps using the binomial formula to calculate the probabilities for parts b), c), and d). If you have a calculator or software that can calculate binomial probabilities, this will be quicker. Otherwise, you can calculate each term manually and sum them up as needed.