[Math] Probability of a target

probabilitystatistics

Suppose the probability of hitting a target is $1/4$. If eight shots are fired at the target, what is the probability that the target is hit at least twice.

I am learning this topic by myself. I am not sure how to solve this problem. I suppose that we would subtract it from one. Here is the work.

$$P(X \geq 2) = 1-P(X < 2)$$

$$ = 1- \binom{8}{0} (0.25)^{0} (0.75)^{8}-\binom{8}{1} (0.25)^{1}(0.75)^{7}$$

I was wondering is this correct. Can you explain if there is an easier way please? Thanks.

Best Answer

Your approach is fine, except that the probability of missing is $0.75$, so your $(1)$'s should be $0.75$'s and you lost a $5$ in the last term-the $0.2$ should be $0.25$

Related Solutions

[Math] Hitting a Target

One way to check theory is to simulate the problem. The code snippet below (using Octave) gives the empirical value of $0.62980$ which is comparable to the answer you have: $0.62419$ (See computation at Wolfram Alpha).

Of course, simulations like the one below are no guarantee that your theoretical answer is correct. However, at the very least they can help rule out incorrect answers and serve as a secondary check to your theoretical analysis.

function ev = ProbEst()

# Probability of hit is 1/5
probHit = 1/5;

totalIter = 10000;
noShots = 10;

# Counter to track if the event of interest
# has occurred 
eventOccurred = 0;

# Set seed of random number generator
rand("state",[11;22;3209;42038021;11]);

for iter =1:totalIter
    targetHit = 0;
    for shot = 1:noShots
        if (rand(1,1) < probHit)
            targetHit += 1;
        endif
    endfor 
    if (targetHit >= 2)  
        eventOccurred += 1;
    endif
endfor

# Output empirical probability to console
eventOccurred/totalIter

endfunction

Probability – Probability That a Man Will Hit the Target

Your first calculation finds the probability that the person hits the target $4$ times in a row. That is very different (and much smaller) than the probability that the person hits at least once.

Let's do the problem in another way, much too long, but it will tell us what is going on.

What is the probability of hitting at least once? The event "at least one hit" could happen in several ways: (i) exactly $1$ hit; (ii) exactly $2$ hits; (iii) exactly $3$ hits; (iv) exactly $4$ hits.

(i) The probability of exactly one hit is $\binom{4}{1}(1/4)(3/4)^3$. This is because the hit could happen in any one of $4$ (that is, $\binom{4}{1}$) places. Write H for hit and M for miss. The probability of the pattern HMMM is $(1/4)(3/4)(3/4)(3/4)$. Similarly, the probability of MHMM is $(3/4)(1/4)(3/4)(3/4)$. You will notice this probability is the same as the probability of HMMM. We get the same probability for MMHM and for MMMH, for our total of $\binom{4}{1}(1/4)(3/4)^3$.

(ii) Similarly, the probability of exactly $2$ hits is $\binom{4}{2}(1/4)^2(3/4)^2$.

(iii) The probability of $3$ hits is $\binom{4}{3}(1/4)^3(3/4)$.

(iv) The probability of $4$ hits is $\binom{4}{4}(1/4)^4$. This is the $(1/4)^4$ that you calculated.

Add up. We get the required answer.

However, that approach is a lot of work. It is much easier to find the probability of no hits, which is the probability of getting MMMM. This is $(3/4)^4$. So the probability that the event "at least one hit" doesn't happen is $(3/4)^4$. So the probability that the event "at least one hit" does happen is $1-(3/4)^4$.

Best Answer

Related Solutions

[Math] Hitting a Target

Probability – Probability That a Man Will Hit the Target

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