Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Car
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 323 times and lost 677 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Car
11. Goat
12. Goat
13. Goat
14. Car
15. Car
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Car
24. Car
25. Goat
26. Goat
27. Goat
28. Goat
29. Car
30. Car
31. Goat
32. Goat
33. Car
34. Car
35. Car
36. Goat
37. Goat
38. Goat
39. Car
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Car
47. Goat
48. Car
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Car
56. Car
57. Car
58. Goat
59. Goat
60. Goat
61. Goat
62. Car
63. Goat
64. Goat
65. Car
66. Goat
67. Goat
68. Goat
69. Car
70. Goat
71. Car
72. Goat
73. Goat
74. Car
75. Car
76. Car
77. Car
78. Goat
79. Goat
80. Car
81. Car
82. Goat
83. Car
84. Goat
85. Goat
86. Car
87. Car
88. Goat
89. Car
90. Goat
91. Goat
92. Goat
93. Car
94. Car
95. Car
96. Goat
97. Goat
98. Car
99. Car
100. Goat
101. Goat
102. Goat
103. Goat
104. Car
105. Goat
106. Goat
107. Car
108. Goat
109. Goat
110. Car
111. Goat
112. Goat
113. Car
114. Goat
115. Goat
116. Goat
117. Goat
118. Car
119. Car
120. Goat
121. Goat
122. Car
123. Goat
124. Goat
125. Car
126. Goat
127. Car
128. Goat
129. Goat
130. Car
131. Goat
132. Car
133. Car
134. Car
135. Car
136. Goat
137. Goat
138. Goat
139. Car
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Goat
147. Car
148. Goat
149. Car
150. Goat
151. Car
152. Goat
153. Goat
154. Goat
155. Car
156. Goat
157. Goat
158. Car
159. Car
160. Goat
161. Goat
162. Goat
163. Goat
164. Car
165. Goat
166. Car
167. Goat
168. Goat
169. Goat
170. Goat
171. Car
172. Car
173. Goat
174. Goat
175. Car
176. Car
177. Goat
178. Goat
179. Goat
180. Goat
181. Goat
182. Goat
183. Car
184. Goat
185. Goat
186. Goat
187. Goat
188. Car
189. Goat
190. Car
191. Goat
192. Goat
193. Goat
194. Car
195. Goat
196. Goat
197. Goat
198. Goat
199. Goat
200. Goat
201. Goat
202. Car
203. Goat
204. Car
205. Goat
206. Car
207. Car
208. Goat
209. Goat
210. Car
211. Goat
212. Goat
213. Car
214. Car
215. Goat
216. Car
217. Car
218. Car
219. Goat
220. Goat
221. Car
222. Goat
223. Car
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Goat
230. Goat
231. Goat
232. Car
233. Goat
234. Goat
235. Goat
236. Goat
237. Goat
238. Goat
239. Car
240. Goat
241. Goat
242. Car
243. Goat
244. Goat
245. Car
246. Goat
247. Goat
248. Car
249. Car
250. Goat
251. Goat
252. Goat
253. Goat
254. Car
255. Goat
256. Goat
257. Car
258. Car
259. Goat
260. Goat
261. Goat
262. Car
263. Goat
264. Goat
265. Goat
266. Goat
267. Goat
268. Goat
269. Goat
270. Goat
271. Goat
272. Goat
273. Car
274. Goat
275. Goat
276. Car
277. Car
278. Goat
279. Goat
280. Goat
281. Goat
282. Goat
283. Goat
284. Goat
285. Goat
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Car
292. Goat
293. Car
294. Goat
295. Goat
296. Goat
297. Goat
298. Goat
299. Goat
300. Goat
301. Goat
302. Goat
303. Goat
304. Goat
305. Goat
306. Goat
307. Car
308. Car
309. Goat
310. Goat
311. Car
312. Goat
313. Goat
314. Goat
315. Goat
316. Goat
317. Car
318. Car
319. Car
320. Goat
321. Car
322. Goat
323. Car
324. Goat
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Car
331. Car
332. Goat
333. Car
334. Goat
335. Car
336. Goat
337. Car
338. Goat
339. Goat
340. Car
341. Goat
342. Car
343. Goat
344. Goat
345. Goat
346. Goat
347. Goat
348. Goat
349. Goat
350. Goat
351. Goat
352. Car
353. Car
354. Goat
355. Goat
356. Goat
357. Car
358. Car
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Car
365. Car
366. Car
367. Goat
368. Goat
369. Car
370. Goat
371. Goat
372. Goat
373. Goat
374. Goat
375. Goat
376. Car
377. Car
378. Goat
379. Goat
380. Car
381. Car
382. Goat
383. Goat
384. Car
385. Car
386. Car
387. Goat
388. Goat
389. Goat
390. Car
391. Car
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Car
401. Goat
402. Goat
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Goat
409. Goat
410. Car
411. Car
412. Goat
413. Goat
414. Goat
415. Car
416. Car
417. Goat
418. Goat
419. Car
420. Car
421. Goat
422. Goat
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Car
437. Car
438. Goat
439. Car
440. Goat
441. Car
442. Goat
443. Goat
444. Car
445. Goat
446. Goat
447. Car
448. Goat
449. Car
450. Car
451. Car
452. Goat
453. Goat
454. Car
455. Goat
456. Goat
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Goat
463. Goat
464. Goat
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Goat
471. Goat
472. Car
473. Goat
474. Goat
475. Car
476. Car
477. Goat
478. Goat
479. Goat
480. Goat
481. Car
482. Goat
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Car
492. Car
493. Car
494. Car
495. Goat
496. Goat
497. Car
498. Goat
499. Goat
500. Car
501. Car
502. Car
503. Goat
504. Goat
505. Car
506. Goat
507. Car
508. Goat
509. Goat
510. Goat
511. Car
512. Goat
513. Car
514. Goat
515. Goat
516. Car
517. Goat
518. Car
519. Goat
520. Goat
521. Goat
522. Car
523. Goat
524. Goat
525. Car
526. Goat
527. Goat
528. Goat
529. Goat
530. Car
531. Goat
532. Goat
533. Car
534. Car
535. Car
536. Car
537. Goat
538. Goat
539. Car
540. Car
541. Car
542. Goat
543. Goat
544. Goat
545. Car
546. Goat
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Goat
553. Goat
554. Car
555. Goat
556. Goat
557. Goat
558. Goat
559. Car
560. Car
561. Goat
562. Goat
563. Car
564. Car
565. Goat
566. Goat
567. Car
568. Goat
569. Goat
570. Car
571. Goat
572. Car
573. Goat
574. Goat
575. Car
576. Goat
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Car
583. Car
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Goat
592. Car
593. Car
594. Car
595. Goat
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Car
602. Goat
603. Goat
604. Car
605. Car
606. Goat
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Goat
614. Car
615. Goat
616. Goat
617. Goat
618. Car
619. Car
620. Car
621. Car
622. Goat
623. Car
624. Goat
625. Car
626. Goat
627. Car
628. Goat
629. Goat
630. Car
631. Car
632. Car
633. Goat
634. Car
635. Goat
636. Car
637. Car
638. Goat
639. Car
640. Goat
641. Car
642. Goat
643. Goat
644. Car
645. Goat
646. Goat
647. Goat
648. Car
649. Goat
650. Goat
651. Goat
652. Car
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Car
659. Goat
660. Car
661. Car
662. Goat
663. Goat
664. Car
665. Car
666. Goat
667. Car
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Goat
674. Goat
675. Goat
676. Car
677. Goat
678. Car
679. Car
680. Goat
681. Car
682. Goat
683. Car
684. Goat
685. Car
686. Goat
687. Car
688. Car
689. Goat
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Car
697. Goat
698. Car
699. Goat
700. Car
701. Goat
702. Goat
703. Goat
704. Goat
705. Goat
706. Goat
707. Car
708. Car
709. Goat
710. Goat
711. Goat
712. Car
713. Car
714. Car
715. Goat
716. Goat
717. Goat
718. Goat
719. Goat
720. Car
721. Goat
722. Goat
723. Car
724. Goat
725. Car
726. Goat
727. Goat
728. Goat
729. Goat
730. Car
731. Car
732. Goat
733. Car
734. Goat
735. Goat
736. Goat
737. Car
738. Goat
739. Car
740. Car
741. Goat
742. Car
743. Goat
744. Goat
745. Car
746. Goat
747. Car
748. Goat
749. Car
750. Car
751. Goat
752. Goat
753. Car
754. Goat
755. Goat
756. Goat
757. Goat
758. Goat
759. Goat
760. Car
761. Goat
762. Goat
763. Car
764. Goat
765. Goat
766. Goat
767. Goat
768. Goat
769. Goat
770. Car
771. Goat
772. Goat
773. Goat
774. Goat
775. Goat
776. Goat
777. Goat
778. Goat
779. Goat
780. Car
781. Car
782. Goat
783. Car
784. Car
785. Goat
786. Goat
787. Goat
788. Car
789. Goat
790. Goat
791. Goat
792. Goat
793. Car
794. Goat
795. Goat
796. Goat
797. Goat
798. Goat
799. Goat
800. Goat
801. Car
802. Goat
803. Car
804. Car
805. Goat
806. Goat
807. Goat
808. Goat
809. Goat
810. Goat
811. Car
812. Goat
813. Goat
814. Car
815. Goat
816. Goat
817. Goat
818. Goat
819. Car
820. Car
821. Goat
822. Car
823. Goat
824. Goat
825. Goat
826. Goat
827. Goat
828. Car
829. Car
830. Car
831. Car
832. Car
833. Goat
834. Goat
835. Goat
836. Goat
837. Goat
838. Goat
839. Goat
840. Car
841. Goat
842. Goat
843. Car
844. Car
845. Goat
846. Goat
847. Goat
848. Car
849. Goat
850. Goat
851. Goat
852. Car
853. Car
854. Goat
855. Goat
856. Car
857. Car
858. Goat
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Car
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Car
872. Goat
873. Goat
874. Car
875. Goat
876. Goat
877. Goat
878. Goat
879. Car
880. Car
881. Car
882. Goat
883. Goat
884. Goat
885. Goat
886. Car
887. Goat
888. Goat
889. Goat
890. Goat
891. Goat
892. Car
893. Car
894. Goat
895. Goat
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Car
903. Car
904. Goat
905. Goat
906. Goat
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Car
916. Car
917. Goat
918. Car
919. Goat
920. Goat
921. Goat
922. Goat
923. Goat
924. Car
925. Car
926. Goat
927. Car
928. Goat
929. Car
930. Goat
931. Car
932. Goat
933. Goat
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Car
940. Goat
941. Goat
942. Goat
943. Goat
944. Car
945. Car
946. Goat
947. Goat
948. Goat
949. Car
950. Goat
951. Goat
952. Goat
953. Goat
954. Car
955. Goat
956. Goat
957. Goat
958. Goat
959. Car
960. Goat
961. Goat
962. Goat
963. Goat
964. Car
965. Goat
966. Goat
967. Car
968. Goat
969. Goat
970. Car
971. Goat
972. Goat
973. Car
974. Goat
975. Goat
976. Car
977. Goat
978. Goat
979. Goat
980. Goat
981. Car
982. Goat
983. Car
984. Car
985. Goat
986. Car
987. Goat
988. Car
989. Car
990. Goat
991. Goat
992. Goat
993. Car
994. Car
995. Car
996. Goat
997. Car
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 102 times and lost 898 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 37 times and lost 63 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}