If the probability of running out of gas is .03 and the probability the electronic starting system will not work is .01

a.) what is the probability there will be enough gas and that the starting system will work? assume the two events are independent

b.) when may independence be a poor assumption?

a. .97*.99 = ?

b. When one variable effects another.

To calculate the probability of two independent events occurring, you simply multiply their individual probabilities together.

a) The probability there will be enough gas is 0.97 (1 - 0.03) since the chance of not running out of gas is 1 minus the probability of running out of gas. The probability that the starting system will work is 0.99 (1 - 0.01) as given. Now, you can multiply these two probabilities:

Probability (enough gas and starting system working) = Probability (enough gas) * Probability (starting system working)
= 0.97 * 0.99
= 0.9603

Therefore, the probability that there will be enough gas and the starting system will work is 0.9603 or approximately 96.03%.

b) Independence may be a poor assumption under certain conditions. Some situations where independence may not hold include:

1. Conditional dependence: When the occurrence of one event affects the likelihood of the other event. For example, if the probability of rain affects the probability of carrying an umbrella, these events are no longer independent.

2. Time dependence: When events occur sequentially and the outcome of one event affects the outcome of the other. For instance, if you withdraw money from an ATM twice in a row without replenishing it, the probability of running out of money increases with each withdrawal.

3. Interactions: When there are interactions or dependencies between variables that impact the probability of events occurring together. For instance, if the probability of getting a job offer is dependent on both your skills and interview performance, these events are not independent.

It is essential to carefully consider the context and relationship between events before assuming independence.