A game show contestant is offered the option of receiving a computer system worth $\$ 2100$ or accepting a chance to win either a luxury vacation worth $\$ 4000$ or a boat worth $\$ 6500$. If the second option is chosen the contestant's probabilities of winning the vacation or the boat are 0.20 and 0.15 , respectively. If the contestant were to turn down the computer system and go for one of the other prizes, what would be the expected winnings?
The expected winnings are $\$ \square$.
Answer
Final Answer: The expected winnings are \(\boxed{\$1775}\).
Step 1 ：A game show contestant is offered the option of receiving a computer system worth \$2100 or accepting a chance to win either a luxury vacation worth \$4000 or a boat worth \$6500. If the second option is chosen the contestant's probabilities of winning the vacation or the boat are 0.20 and 0.15, respectively.
Step 2 ：If the contestant were to turn down the computer system and go for one of the other prizes, we need to calculate the expected winnings.
Step 3 ：The expected value of a random variable is the sum of the possible values each multiplied by the probability of its occurrence. In this case, the random variable is the winnings from the game show, and the possible values are \$4000 and \$6500, with probabilities 0.20 and 0.15 respectively.
Step 4 ：The expected winnings can be calculated as follows: Expected winnings = (0.20 * \$4000) + (0.15 * \$6500)
Step 5 ：The expected winnings from the second option is \$1775. However, this is less than the value of the computer system, which is \$2100. Therefore, the contestant should choose the computer system.
Step 6 ：Final Answer: The expected winnings are \(\boxed{\$1775}\).
