# Math equation could help find missing Malaysian plane

Bayes’ Theorem helped researchers locate Air France Flight 447’s black box in 2011

(Video: How Bayesian Search found the USS Scorpion)

Days after a Malaysian airliner with 239 people aboard went missing en route to Beijing, searchers are still struggling to find any confirmed sign of the plane. Authorities have acknowledged that they didn’t even know what direction it was heading when it disappeared.

As frustrations mount over the failures of the latest technology in the hunt for Malaysia Airlines Flight MH370, some scientists say an 18th-century mathematical equation – used in a previous search for an Air France jetliner’s black box recorder – could help pinpoint the location of the Malaysian plane.

Indonesian Air Force officers examine a map of the Malacca Strait during a briefing following a search operation for the missing Malaysia Airlines Boeing 777, at Suwondo air base in North Sumatra, Indonesia, on Wednesday.AP2014

In 2009, Air France Flight 447 en route to Paris from Rio de Janeiro vanished over the Atlantic Ocean, triggering the most expensive and exhaustive search effort ever conducted for a plane. After two years, officials could only narrow the location of the plane’s black box down to an area the size of Switzerland.

But Flight 447’s black box was found in just five days after authorities contacted scientific consultants who applied a centuries-old equation called Bayes’ Theorem.

## What is Bayes’ Theorem

Mathematically, Bayes’ theorem gives the relationship between the probabilities of A and B, P(A) and P(B), and the conditional probabilities of A given B and B given A, P(A|B) and P(B|A). In its most common form, it is: (Wikipedia)

$\displaystyle\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}$

## Proof of Bayes’ theorem (Theorem useful for finding MH370 plane)

The proof of Bayes’ theorem is actually relatively simple, the only requirement is to know the formula for conditional probability (Learnt in H1/H2 Maths): $\displaystyle \boxed{P(A|B)=\frac{P(A\cap B)}{P(B)}}$

From this, we have $\displaystyle \boxed{P(A\cap B)=P(A|B)P(B)}$

Similarly, $\displaystyle \boxed{P(B\cap A)=P(B|A)P(A)}$

But since $\displaystyle P(A\cap B)=P(B\cap A)$, we have $P(A|B)P(B)=P(B|A)P(A)$. Dividing throughout by $P(B)$ gives Bayes’ Formula: $\displaystyle\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}$

Sincerely wishing that the MH370 plane will be found soon, and hopefully the passengers are still alive.

Also see: Bayesian search theory (Bayesian search theory is the application of Bayesian statistics to the search for lost objects. It has been used several times to find lost sea vessels, for example the USS Scorpion. It also played a key role in the recovery of the flight recorders in the Air France Flight 447 disaster of 2009.)