Doubling down. recall the dice game played between alice and bob: alice rolls two dice, and tells bob the sum; then bob guesses a number from 1 to 6. bob wins the game if his guess appears on either of the two dice. now suppose alice and bob play for money. if bob wins, he wins a dollar, and if he loses, alice wins a dollar. we can model this using a random variable w representing bob’s winnings. we saw that using an optimal strategy, bob wins with probability 5/9. this means eii|-(1)(5/9)+(-1)(4/9)-1/9to make the game more interesting, alice allows bob to "double down": this means that after alice tells bob the sum, bob is allowed to double his bet from one dollar to two dollars, if he so chooses. give an optimal strategy for bob (for both his guess and the double-down decision) and compute e[w] for this strategy. hint: if z is a random variable representing the sum of the two dice, compute e[w] using the law of total expectation:

21

step-by-step explanation:

11279

a) one student is chosen from classroom a, then a student is chosen from classroom b.  

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b.   this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes.   this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes.   this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes.   this makes them dependent events.

this is answer of 144 cubic meters

on monday rodney ate 1/4th of the sandwich = 3/12

on tuesday he ate 1/3rd of the sandwich = 4/12

therefore in total he has eaten = 3/12 + 4/12 = 7/12

thus we can say that he has eaten 7/12 of the sandwich on monday and tuesday alltogether.

happy to

pls mark as brainliest.