How to interpret P(X>x) in a Poisson distribution question

Question

Cotton cloth is sold from long rolls of cloth. The number of flaws on a randomly chosen piece of cloth of length $a$ meters has a Poisson distribution with mean $0.8a$. The random variable $X$ is the length of cloth, in metres, between two successive flaws. For $x \ge 0$, find $P(X>x).$

Source : Cambridge A-level Further Mathematics 9231_s15_qp_21 Question 9

My work

$W:$ number of flaws

Rate of flaws = 0.8 flaws per metre

$W \sim Poisson(0.8)$

$X:$ distance between 2 successive flaws

$X$ is exponentially distributed.

PDF of X : $f(w)=0.8e^{-0.8w}$ for $w \ge 0$

$P(X>x)=\int^{\infty}_x 0.8e^{-0.8w}dw=e^{-0.8x}$

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In the mark scheme of this question it is simply written :

$P(X>x)= P($zero flaws in $x$ m$)=P_0(0.8x)=e^{-0.8x}$

I don't understand why $P(X>x)=$ $P($zero flaws in $x$ m$)$

Answer 1

Intuitively when you defined the variable $X$ you did not specified if it was between the time of the second and third flaw, third and fourth,... . Because it does not affect the distribution of $X$ (see lack of memory for exponential distributions). Therefore when you look at the event ''$X>x$'' you just want to be zero flaw in x metters. .