Austin and Betty are entrepreneurs with many ideas. They need initial investment. A venture capitalist Craig decides to invest one million dollars in a new business. One at a time, Austin and Betty ask Craig for the initial investment in a new business.

Austin starts first and he has 25% chance to convince Craig in an attempt. Austin and Betty have equal chances to get $1,000,000 at the end.

What is Betty’s probability to get $1,000,000 in a single attempt?

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Real examples help to better understand how to interpret the statistics.

False Positive Paradox: A particular medical test for a disease is 96% accurate. If one has the disease, the test comes back ‘Yes’ 96% of the time, and if one does not have the disease, the test comes back ‘Yes’ 4% of the time.

If 100 of 10000 tested patients have the disease, what is the probability that the person with the diagnosis ‘Yes’ has the disease?

Effective Average Infection Ratio: R is the effective average infection ratio for a disease, also known as the reproduction number. It is the average number of secondary infections caused by one person. (Infections caused by the secondary infections – which would be tertiary infections – are not counted). Consider 50 infected people. Suppose 49 spread the infection to nobody, but one person spreads the infection to 60 people.

The Pfizer COVID-19 vaccine-candidate interim results from 8 Nov 2020 showed 38,955 participants in a placebo controlled double-blind trial. 94 participants became “evaluable”, which we presume to mean they showed COVID-19 symptoms. The analysis presented was that the vaccine efficacy rate was above 90%. What is the maximum number of (genuinely) vaccinated people who showed COVID-19 symptoms?

In the UK, December 2020, up to 2 million university students were potentially going home for Xmas during the middle of the COVID-19 pandemic. A new fast COVID-19 test had been developed to spot the causative virus, SARS-CoV-2. These lateral flow tests had the characteristics shown in the image. At the time, around 1% of the population had the virus within the community.

The scientific advice, given on prime-time news channels, was that a pair of negative tests meant it was safe to go home, as a negative test meant a 99.75% chance of not having the virus.

There are 5 counters in a bag. Three are Argentinean (blue) and two are Brazilian (green). Three counters are randomly picked out of the bag, one by one. They are not returned to the bag.

What probability is higher?

A. The probability of choosing three Argentinean counters in a row.

B. The probability of choosing two Brazilian counters in a row and then one Argentinean counter.

There was a tradition in old Russia. A would-be bride gathers six long pieces of straw and grasp them in her hand. She then randomly ties pairs of knots on the top and the bottom. Since there are six blades of grass sticking out above and below the hand, she will tie three knots on the top and three knots on the bottom. If she forms one big ring, she gets married soon.

Estimate the probability that the girl will get married soon.