Probability of intersection between UNRWA staff and Al-Qassam fighters in the Gaza Strip

I was reading Richard Stallman’s political posts on his website. Regarding accusations by the Israeli government against UNRWA, he briefly commented on what he sees as a rigorous hiring process by UNRWA for excluding armed Hamas members:

UNRWA investigated accusations that 19 of its Gaza staff were involved in HAMAS’s Oct 7 terrorist attack. For 9 of them it found sufficient evidence to fire them.

Since UNRWA has 13,000 employees in Gaza, to be that successful in excluding HAMAS militants from its work force is good going. If we estimate roughly a million adult men in Gaza, 13,000 is roughly 1% of them. I think that HAMAS militants numbered around 10,000, also roughly 1% of the total. So the intersection of the two sets, by pure chance, would have numbered around 100.

To have had only a tenth that many indicates an effective effort by UNRWA to avoid hiring HAMAS militants.

It is absurd to fault UNRWA for not doing even better than that!

The US should resume its support for UNRWA.

— Richard Stallman¹

I thought it’s quite interesting since it’s a simple thought experiment that never dawned upon me. So here is a more verbose version:

Asumptions:

• 15,000 males in Gaza had been members of the Al-Qassam Brigade (certain estimates — such as those from the US or Israel — are considerably higher; but we’ll take 15,000)
• Number of non-male Al-Qassam members is negligible
• Gaza’s male population was ~1.10 million²
• UNRWA had ~13,000 local staff in the Gaza Strip
• Let gender ratio among the staff be equal
• Being an UNRWA staff and an Al-Qassam member are independent events

Calculations:

• Proportion of male population who were Al-Qassam members: \frac{15,000}{1.10*10^6}\approx 1.364\%
• Proportion of male population who were UNRWA staff: \frac{13,000*0.5}{1.10*10^6}\approx 0.591\%

Apply the general multiplication rule \Pr(A\cap B)=\Pr(A)\Pr(B):

\begin{aligned}\Pr(\text{Al-Qassam}\cap \text{UNRWA})&=\Pr(\text{Al-Qassam})\Pr(\text{UNRWA}) \\ &=1.364\%*0.591\% \\ &\approx 0.0081\%\end{aligned}

Therefore, in such a uniform sample space, the number of males who were simultaneously UNRWA staff and Al-Qassam members would be 0.0081\% *1.10*10^6\approx 89.

1. https://www.stallman.org/archives/2024-may-aug.html. Section “7 August 2024 (Conclusions in UNRWA-HAMAS ties)” ↩︎
2. https://www.pcbs.gov.ps/portals/_pcbs/PressRelease/Press_En_InterPopDay2022E.pdf ↩︎