I'm preparing for a test and one of the questions in my text is as follows:
In a trivia contest,Sam has drawn a topic he knows nothing about, so he makes random guesses for the 10 T/F questions. Use the binomial theorem to help find
a)the number of ways Sam can answer the test using exactly four trues
b)the number of ways that Sam can answer the test using at least one true. The answers are 210 and 1023. I have no idea how to approach this.
The binomial theorem can be applied to the 10 true-false questions as follows:
Let (T+F)^10 represent the choices Sam could make, i.e. for each question, he enters either true or false.
We know that:
(T+F)^10=
T^10+10*F*T^9+45*F^2*T^8+120*F^3*T^7+210*F^4*T^6+252*F^5*T^5+210*F^6*T^4+120*F^7*T^3+45*F^8*T^2+10*F^9*
T+F^10
The coefficients tell us the number of ways the combinations of true/false can be made.
For example:
There is only one way to write 10 trues (obviously).
But there are 10 ways to write 1 false and 9 trues (a false at each of the 10 questions), and so on.
The coefficients are obtained, fortunately, easily by the formula for "n choose r", or C(n,r), or n!/(r!(n-r)!).
So for 4 trues and 6 falses, the number of ways is C(10,4)=10*9*8*7/(1*2*3*4)=210.
Similarly, to have at least one true, we can do a summation of C(10,1),C(10,2)....C(10,10) to get 1023.
Alternatively, we note that there are 2^10=1024 ways to do the exam, out of which only one case (10 false) does not have at least one true. So the number of ways for at least one true is 1024-1=1023.
Thank you very much!
You're welcome!
(10/4) computed in binomial expansion? that's ten over four in parenthesis.
To answer these questions, we can use the binomial theorem which is a formula that allows us to calculate the probabilities of different outcomes in a given situation.
Before we dive into the calculations, let's first understand the problem. Sam has to answer 10 True/False (T/F) questions. Since Sam knows nothing about the topic, he randomly guesses the answers. We need to use the binomial theorem to find the number of ways Sam can answer the test with certain conditions.
a) Number of ways Sam can answer the test using exactly four trues:
To solve this, we need to find the probability of getting exactly four true answers out of ten. The formula for the binomial distribution is:
P(X = k) = nCk * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- nCk is the combination formula for choosing k items out of n,
- p is the probability of success,
- (1-p) is the probability of failure,
- k is the number of successes, and
- n is the total number of trials.
In this case, "k" represents the number of true answers, which is 4. The probability of getting a true answer is 0.5 (since Sam randomly guesses). The probability of failure (getting a false answer) is also 0.5. The total number of trials is 10.
Using these values in the formula, we can calculate:
P(X = 4) = 10C4 * 0.5^4 * 0.5^6
= 210 * 0.0625 * 0.015625
= 210 * 0.0009765625
= 0.205078125
So, the number of ways Sam can answer the test with exactly four trues is 210.
b) Number of ways Sam can answer the test using at least one true:
To solve this, we need to find the probability of getting at least one true answer out of ten. This can be calculated by subtracting the probability of getting no true answers (all false) from the total probability space (all possible outcomes).
The probability of getting no true answers is calculated as follows:
P(X = 0) = 10C0 * 0.5^0 * 0.5^10
= 1 * 1 * 0.0009765625
= 0.0009765625
To find the probability of at least one true answer, we subtract this probability from 1 (the total probability space):
P(at least one true) = 1 - P(X = 0)
= 1 - 0.0009765625
= 0.9990234375
Now, we need to calculate the number of ways to achieve this probability. Since Sam can make independent choices for each question, we can multiply the probability by the total number of outcomes:
Number of ways = P(at least one true) * Total number of outcomes
= 0.9990234375 * 2^10
= 0.9990234375 * 1024
≈ 1023
So, the number of ways Sam can answer the test with at least one true is approximately 1023.
By using the binomial theorem and these calculations, we arrive at the answers provided in your question - 210 and 1023.