4 Kareem and Sam play a game in which each holds a hand of three cards.

• Kareem's cards are numbered 1, 2 and 5.
• Sam's cards are numbered 3, 4 and 6 .

In each round Kareem and Sam simultaneously choose a card from their hand, they show their chosen card to the other player and then return the card to their own hand.

• If the sum of the numbers on the cards shown is even then the number of points that Kareem scores is \(2 k\), where \(k\) is the number on Kareem's card.
• If the sum of the numbers on the cards shown is odd then the number of points that Kareem scores is \(4 - s\), where \(s\) is the number on Sam's card.
  1. Complete the pay-off matrix for this game, to show the points scored by Kareem.
  2. Write down which card Kareem should play to maximise the number of points that he scores for each of Sam’s choices.
  3. Determine the play-safe strategy for Kareem.
  4. Explain why Kareem should never play the card numbered 1.

Sam chooses a card at random, so each of Sam’s three cards is equally likely.

Calculate Kareem's expected score for each of his remaining choices.