Tree Diagram and Probability with Sweets

Jenny has a bag with seven blue sweets and three red sweets in it. She picks a sweet at random from the bag, replaces it, and then picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks:

a) two red sweets,

b) no red sweets,

c) at least one blue sweet,

d) one sweet of each color.

This problem is an excellent exploration of basic probability concepts using a classic scenario that involves repeated trials with replacement. The problem invites you to draw a tree diagram, which is a powerful tool to visually represent all possible outcomes of a probabilistic scenario and their associated probabilities. Tree diagrams can help simplify your analysis by clearly laying out the sequence of events and outcomes. In this problem, each level of the tree represents one pick of a sweet, and each branch shows the probability of picking either a red or blue sweet. By calculating the entire tree, you'll be able to sum probabilities of different branches to answer the given questions about picking sweets by color.

The first question asks you to calculate the probability of picking two red sweets. By identifying the branch that corresponds to picking a red sweet twice and calculating the probability along that path, you reinforce the concept of independent events and multiplication of probabilities across independent events. For the probability of picking no red sweets, you need to identify the sequence where only blue sweets are picked. This will also demonstrate how probabilities of certain complementary events can be summed to make calculations easier.

Finally, considering the questions about picking at least one blue sweet and one sweet of each color, you will engage with the concepts of complementary events and combined outcomes. "At least one" scenarios often require you to calculate the probability of the opposite event (none of the desired outcome) and subtract it from one. For the one of each color scenario, you'll use branches that represent one red and one blue sweet, including both possible sequences. By solving these questions, you not only practice tree diagram usage but also deepen your understanding of probability in experiments with replacement.