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 7 times and lost 3 times.

1. Car
2. Car
3. Car
4. Car
5. Goat
6. Car
7. Goat
8. Car
9. Car
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 327 times and lost 673 times.

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