Question

A junk box in your room contains a dozen old batteries, five of which are totally dead. You start picking batteries one at a time and testing them. Find the probability of each outcome. a) The first two you choose are both good. b) At least one of the first three works. c) The first four you pick all work. d) You have to pick 5 batteries to find one that works.

Short answer

a) \(\frac{7}{22}\), b) \(\frac{21}{22}\), c) \(\frac{7}{99}\), d) \(\frac{1}{113}\).

Step-by-step solution

Determine Total and Good Batteries

You have a total of 12 batteries in the box. Out of these, 7 batteries are good since 5 are totally dead.

Calculate Probability for Part (a)

a) To find the probability that the first two batteries are both good, use the formula for probability of consecutive events: \[\text{P(First Two Good)} = \frac{\text{Number of Good Batteries}}{\text{Total Batteries}} \times \frac{\text{Number of Remaining Good Batteries}}{\text{Total Remaining Batteries}}\]Thus, \[\text{P(First Two Good)} = \frac{7}{12} \times \frac{6}{11} = \frac{42}{132} = \frac{7}{22}\].

Calculate Probability for Part (b)

b) Calculate the probability that at least one of the first three batteries works. Use the complementary probability for this case: it is easier to find the probability that none of the three work, and subtract that from 1. The probability that each one picked is dead is \[\text{P(All Dead)} = \frac{5}{12} \times \frac{4}{11} \times \frac{3}{10} = \frac{60}{1320} = \frac{1}{22}\].Thus, \[\text{P(At Least One Works) = 1 - P(All Dead)} = 1 - \frac{1}{22} = \frac{21}{22}\].

Calculate Probability for Part (c)

c) To find the probability that the first four batteries you pick all work, use the formula: \[\text{P(All Four Good)} = \frac{7}{12} \times \frac{6}{11} \times \frac{5}{10} \times \frac{4}{9} = \frac{840}{11880} = \frac{7}{99}\].

Calculate Probability for Part (d)

d) The probability that you have to pick 5 batteries to find one that works. You want the first four to be dead and the fifth to be good: \[\text{P(First Four Dead, Fifth Good)} = \frac{5}{12} \times \frac{4}{11} \times \frac{3}{10} \times \frac{2}{9} \times \frac{7}{8} = \frac{840}{95040} = \frac{1}{113}\].

Key concepts

Combinatorics

In probability, combinatorics is the mathematics of counting and arrangements. It's all about figuring out how many different ways you can do something. When dealing with problems involving probability, like picking batteries from a box, combinatorics helps us understand the different possibilities. For example, in the exercise, we have a box of 12 batteries, with 7 of them being good and 5 dead. When we pick two batteries, there are multiple outcomes. We can pick two good ones, two dead ones, or one of each. We use combinatorics to find out exactly how many different ways we can make these selections. This is crucial when calculating probabilities because we need to know the total number of possible outcomes – which is also known as the sample space. Without understanding all the possible outcomes, we cannot compute an accurate probability.

Complementary Probability

Sometimes, it's easier to find the probability of something not happening and then subtract that from 1 to get the desired probability. This is known as finding the complementary probability. It is especially useful when directly calculating the probability of an event is complicated. In the exercise, consider Part (b), where we want to find the probability that at least one of the first three batteries works. Directly finding the probability for this scenario would be more complex. Instead, it's simpler to calculate the probability of the complementary event – in this case, that none of the three batteries work – and subtract this from 1. Using complementary probability simplifies calculations, particularly when dealing with events that have many components or where direct computation is complex. By focusing on what doesn't happen, we can often more easily solve what does happen.

Conditional Probability

Conditional probability comes into play when the likelihood of an event is determined by considering that another event has already occurred. This dependence directly affects how we calculate the probability. In our exercise, when calculating the probability for two batteries both being good, we must consider it a dependent probability. We know the second pick is influenced by the result of the first pick. After removing one battery, the amounts left for both good and total batteries change. This interdependence is crucial in 'conditional probability,' as it determines the likelihood of subsequent events. Understanding how events are connected, and how prior events affect future choices, is vital for getting accurate probability outcomes.